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#### Page No 1.09:

To Prove: that the product of two consecutive integers is divisible by 2.

Proof: Let n − 1 and n be two consecutive positive integers.

Then their product is n (n − 1) = n2n

We know that every positive integer is of the form 2q or 2q + 1 for some integer q.

So let n = 2q

So, n2n = (2q)2 − (2q)

Let n = 2q + 1

So, n2n = (2q + 1)2 − (2q + 1)

Hence it is proved that that the product of two consecutive integers is divisible by 2

#### Page No 1.09:

Given: If a and b are two odd positive integers such that a > b.

To Prove: That one of the two numbers and is odd and the other is even.

Proof: Let a and b be any odd odd positive integer such that a > b.

Since any positive integer is of the form q, 2q + 1

Let a = 2q + 1 and b = 2m + 1, where, q and m are some whole numbers

which is a positive integer.
Also,

Given, a > b

$\therefore$ 2q + 1 > 2m + 1
$⇒$ 2q > 2m
$⇒$ q > m

$\therefore$

Thus, $\frac{\left(a-b\right)}{2}$ is a positive integer.

Now, we need to prove that one of the two numbers $\frac{\left(a+b\right)}{2}$ and$\frac{\left(a-b\right)}{2}$ is odd and other is even.

Consider,
, which is odd positive integer.

Also, we know from the proof above that  $\frac{\left(a+b\right)}{2}$ and$\frac{\left(a-b\right)}{2}$ are positive integers.

We know that the difference of two positive integers is an odd number if one of them is odd and another is even. (Also, difference between two odd and two even integers is even)

Hence it is proved that If a and b are two odd positive integers such that a > b then one of the two numbers and is odd and the other is even.

#### Page No 1.09:

To Prove: that the square of an odd positive integer is of the form 8q + 1, for some integer q.

Proof: Since any positive integer n is of the form 4m + 1 and 4m + 3

If n = 4m + 1

If n = 4m + 3

Hence n2 integer is of the form 8q + 1, for some integer q.

#### Page No 1.09:

To Show: That any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5 where q is any some integer.

Proof: Let a be any odd positive integer and b = 6.

Then, there exists integers q and r such that

a = 6q + r, 0 r < 6 (by division algorithm)

a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4

But 6q or 6q + 2 or 6q + 4 are even positive integers.

Hence it is proved that any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5, where q is any some integer.

#### Page No 1.09:

To Prove: that the square of an positive integer is of the form 3m or 3m + 1 but not of the form 3m + 2.

Proof: Since positive integer n is of the form of 3q, 3q + 1 and 3q + 2

If n = 3q

If n = 3q + 1

Then, n2 = (3q + 1)2

If n = 3q + 2

Then, n2 = (3q + 2)2

Hence n2 integer is of the form 3m, 3m + 1 but not of the form 3m + 2.

#### Page No 1.09:

To Prove: that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.

Proof: Since positive integer n is of the form of 2q or 2q + 1

If n = 2q

If n = 2q + 1

Hence it is proved that the square of any positive integer is of the form 4q or 4q + 1, for some integer q.

#### Page No 1.09:

To Prove: that the square of any positive integer is of the form 5q or 5q + 1, 5q + 4 for some integer q.

Proof: Since positive integer n is of the form of 5q or 5q + 1, 5q + 4

If n = 5q

If n = 5q + 1

If n = 5q + 2

If n = 5q + 4

Hence it is proved that the square of a positive integer is of the form 5q or 5q + 1, 5q + 4 for some integer q.

#### Page No 1.09:

To Prove: that if a positive integer is of the form 6q + 5 then it is of the form 3q + 2 for some integer q, but not conversely.

Proof: Let n = 6q + 5

Since any positive integer n is of the form of 3k or 3k + 1, 3k + 2

If q = 3k

If q = 3k + 1

If q = 3k + 2

Consider here 8 which is the form 3q + 2 i.e. 3 × 2 + 2 but it can’t be written in the form 6q + 5. Hence the converse is not true

#### Page No 1.09:

To Prove: that the square of a positive integer of the form 5q + 1 is of the same form

Proof: Since positive integer n is of the form 5q + 1

If n = 5q + 1

Hence n2 integer is of the form 5m + 1.

#### Page No 1.09:

To Prove: the product of three consecutive positive integers is divisible by 6.

Proof: Let n be any positive integer.

Since any positive integer is of the form 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4, 6q + 5

If n = 6q

, which is divisible by 6

If n = 6q + 1

Which is divisible by 6

If n = 6q + 2

Which is divisible by 6

Similarly we can prove others.

Hence it is proved that the product of three consecutive positive integers is divisible by 6.

#### Page No 1.09:

To Prove: For any positive integer n, n3n is divisible by 6.

Proof: Let n be any positive integer.

Since any positive integer is of the form 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4, 6q + 5

If n = 6q

If n = 6q + 1

If n = 6q + 2

Similarly we can prove others.

Hence it is proved that for any positive integer n, n3n is divisible by 6.

#### Page No 1.22:

(i) We need to find H.C.F. of 32 and 54.

By applying division lemma

54 = 32$×$1 + 22

Since remainder, apply division lemma on 32 and remainder 22

32 = 22$×$1 + 10

Since remainder, apply division lemma on 22 and remainder 10

22 = 10$×$2 + 2

Since remainder, apply division lemma on 10 and remainder 2

10 = 2$×$5 + 0

Therefore, H.C.F. of 32 and 54 is

(ii) We need to find H.C.F. of 18 and 24.

By applying division lemma

Since remainder, apply division lemma on divisor 18 and remainder 6

Therefore, H.C.F. of 18 and 24 is

(iii) We need to find H.C.F. of 70 and 30.

By applying Euclid’s Division lemma

Since remainder, apply division lemma on divisor 30 and remainder 10

Therefore, H.C.F. of 70 and

(iv) We need to find H.C.F. of 56 and 88.

By applying Euclid’s Division lemma

Since remainder, apply division lemma on 56 and remainder 32

Since remainder, apply division lemma on 32 and remainder 24

Since remainder, apply division lemma on 24 and remainder 8

Therefore, H.C.F. of 56 and.

(v) We need to find H.C.F. of 475 and 495.

By applying Euclid’s Division lemma

Since remainder, apply division lemma on 475 and remainder 20

Since remainder, apply division lemma on 20 and remainder 15

Since remainder, apply division lemma on15 and remainder 5

Therefore, H.C.F. of 475 and.

(vi) We need to find H.C.F. of 75 and 243.

By applying Euclid’s Division lemma

Since remainder, apply division lemma on 75 and remainder 18

Since remainder, apply division lemma on divisor 18 and remainder 3

Therefore, H.C.F. of 75 and.

(vii) We need to find H.C.F. of 240 and 6552.

By applying Euclid’s Division lemma

Since remainder, apply division lemma on divisor 240 and remainder 72

Since remainder, apply division lemma on divisor 72 and remainder 24

Therefore, H.C.F. of 240 and.

(viii) We need to find H.C.F. of 155 and 1385.

By applying Euclid’s Division lemma

Since remainder, apply division lemma on divisor 155 and remainder 145

Since remainder, apply division lemma on divisor 145 and remainder 10

Since remainder, apply division lemma on divisor 10 and remainder 5

Therefore, H.C.F. of 155 and.

(ix) We need to find H.C.F. of 100 and 190.

By applying Euclid’s division lemma

Since remainder, apply division lemma on divisor 100 and remainder 90

Since remainder, apply division lemma on divisor 90 and remainder 10

Therefore, H.C.F. of 100 and.

(x) We need to find H.C.F. of 105 and 120.

By applying Euclid’s division lemma

Since remainder, apply division lemma on divisor 105 and remainder 15

Therefore, H.C.F. of 105 and.

#### Page No 1.22:

(i) We need to find the H.C.F. of 963 and 657 and express it as a linear combination of 963 and 657.

By applying Euclid’s division lemma

Since remainder, apply division lemma on divisor 657 and remainder 306

Since remainder, apply division lemma on divisor 306 and remainder 45

Since remainder, apply division lemma on divisor 45 and remainder 36

Since remainder, apply division lemma on divisor 36 and remainder 9

Therefore, H.C.F. = 9.

Now,

(ii) We need to find the H.C.F. of 592 and 252 and express it as a linear combination of 592 and 252.

By applying Euclid’s division lemma

592 = 252×2+88

Since remainder, apply division lemma on divisor 252 and remainder 88

252 = 88×2+76

Since remainder, apply division lemma on divisor 88 and remainder 76

88 = 76×1+12

Since remainder, apply division lemma on divisor 76 and remainder 12

76 = 12×6+4

Since remainder, apply division lemma on divisor 12 and remainder 4

12 = 4×3+0.

Therefore, H.C.F. = 4.

Now,

(iii) We need to find the H.C.F. of 506 and 1155 and express it as a linear combination of 506 and 1155.

By applying Euclid’s division lemma

Since remainder, apply division lemma on divisor 506 and remainder 143

Since remainder, apply division lemma on divisor 143 and remainder 77

Since remainder, apply division lemma on divisor 77 and remainder 66

Since remainder, apply division lemma on divisor 66 and remainder 11

Therefore, H.C.F. = 11.

Now,

(iv) We need to find the H.C.F. of 1288 and 575 and express it as a linear combination of 1288 and 575.

By applying Euclid’s division lemma

Since remainder, apply division lemma on divisor 506 and remainder 143

Since remainder, apply division lemma on divisor 143 and remainder 77

Therefore, H.C.F. = 23.

Now,

#### Page No 1.22:

(i) Given integers are 225 and 135. Clearly 225 > 135. So we will apply Euclid’s division lemma to 225 and 135, we get,

Since the remainder. So we apply the division lemma to the divisor 135 and remainder 90. We get,

Now we apply the division lemma to the new divisor 90 and remainder 45. We get,

The remainder at this stage is 0. So the divisor at this stage is the H.C.F.

So the H.C.F of 225 and 135 is 45

(ii) Given integers are 38220 and 196. Clearly 38220 > 196. So we will apply Euclid’s division lemma to 38220 and 196, we get,

The remainder at this stage is 0. So the divisor at this stage is the H.C.F.

So the H.C.F of 38220 and 196 is 196

(iii) Given integers are 867 and 255. Clearly 867 > 225. So we will apply Euclid’s division lemma to 867 and 225, we get,

Since the remainder. So we apply the division lemma to the divisor 225 and remainder 192. We get,

Now we apply the division lemma to the new divisor 192 and remainder 33. We get,

Now we apply the division lemma to the new divisor 33 and remainder 27. We get,

Now we apply the division lemma to the new divisor 27 and remainder 6. We get,

Now we apply the division lemma to the new divisor 6 and remainder 3. We get,

The remainder at this stage is 0. So the divisor at this stage is the H.C.F.

So the H.C.F of 867 and 255 is 3.

(iv) Given integers are 184, 230 and 276.
Let us first find the HCF of 184 and 230 by Euclid lemma.
Clearly, 230 > 184. So, we will apply Euclid’s division lemma to 230 and 184.

230 = 184 × 1 + 46
Remainder is 46 which is a non-zero number. Now, apply Euclid’s division lemma to 184 and 46.
184 = 46 × 4 + 0
The remainder at this stage is zero. Therefore, 46 is the HCF of 230 and 184.
Now, again use Euclid’s division lemma to find the HCF of 46 and 276.
276  = 46 × 6 + 0

The remainder at this stage is zero. Therefore, 46 is the HCF of 184, 230 and 276.

(v) Given integers are 136, 170 and 255.
Let us first find the HCF of 136, 170 by Euclid lemma.
Clearly, 170 > 136. So, we will apply Euclid’s division lemma to 136 and 170.

170 = 136 × 1 + 34
Remainder is 34 which is a non-zero number. Now, apply Euclid’s division lemma to 136 and 34.
136 = 34 × 4 + 0
The remainder at this stage is zero. Therefore, 34 is the HCF of 136 and 170.
Now, again use Euclid’s division lemma to find the HCF of 34 and 255.
255  = 34 × 7 +17

Remainder is 17 which is a non-zero number. Now, apply Euclid’s division lemma to 34 and 17.
34 = 17 × 2 + 0
The remainder at this stage is zero. Therefore, 17 is the HCF of 136, 170 and 255.

#### Page No 1.22:

We need to express the H.C.F. of 468 and 222 as

Where x, y are integers in two different ways.

Given integers are 468 and 222, where

By applying Euclid’s division lemma, we get

Since the remainder, so apply division lemma on divisor 222 and remainder 24

Since the remainder, so apply division lemma on divisor 24 and remainder 6

We observe that remainder is 0. So the last divisor 6 is the H.C.F. of 468 and 222 from we have

.

#### Page No 1.22:

We need to find m if the H.C.F of 408 and 1032 is expressible in the form.

Given integers are 408 and 1032 where.

By applying Euclid’s division lemma, we get

Since the remainder, so apply division lemma on divisor 408 and remainder 216

Since the remainder, so apply division lemma on divisor 216 and remainder 192

Since the remainder, so apply division lemma on divisor 192 and remainder 24

We observe that remainder is 0. So the last divisor is the H.C.F of 408 and 1032.

Therefore,

#### Page No 1.22:

We need to find x if the H.C.F of 657 and 963 is expressible in the form.

Given integers are 657 and 963.

By applying Euclid’s division lemma, we get

Since the remainder, so apply division lemma on divisor 657 and remainder 306

Since the remainder, so apply division lemma on divisor 306 and remainder 45

Since the remainder, so apply division lemma on divisor 45 and remainder 36

Since the remainder, so apply division lemma on divisor 36 and remainder 9

Therefore, H.C.F. = 9.

Given H.C.F =.

Therefore,

#### Page No 1.22:

We are given that an army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. We need to find the maximum number of columns in which they can march.

Members in army = 616

Members in band = 32.

Therefore,

Maximum number of columns = H.C.F of 616 and 32.

By applying Euclid’s division lemma

Therefore, H.C.F. = 8

Hence, the maximum number of columns in which they can march is.

#### Page No 1.22:

We need to find the largest number which divides 615 and 963 leaving remainder 6 in each case.

The required number when divides 615 and 963, leaves remainder 6, this means and are completely divisible by the number.

Therefore,

The required number = H.C.F. of 609 and 957.

By applying Euclid’s division lemma

Therefore, H.C.F. = 87.

Hence, the required number is.

#### Page No 1.22:

We need to find the greatest number which divides 285 and 1249 leaving remainder 9 and 7 respectively.

The required number when divides 285 and 1249, leaves remainder 9 and 7, this means are completely divisible by the number.

Therefore, the required number = H.C.F. of 276 and 1242.

By applying Euclid’s division lemma

Therefore, H.C.F. = 138

Hence, required number is.

#### Page No 1.22:

We need to find the largest number which exactly divides 280 and 1245 leaving remainders 4 and 3, respectively.

The required number when divides 280 and 1245, leaves remainder 4 and 3, this means are completely divisible by the number.

Therefore, the required number = H.C.F. of 276 and 1242.

By applying Euclid’s division lemma

Therefore, H.C.F. = 138.

Hence, the required number is.

#### Page No 1.22:

We need to find the largest number that divides 626, 3127 and 15628 and leaves remainders of 1, 2 and 3 respectively.

The required number when divides 626, 3127 and 15628 leaves remainders 1, 2 and 3 this means are completely divisible by the number.

Therefore, the required number = H.C.F. of 625, 3125 and 15625.

First we consider 625 and 3125.

By applying Euclid’s division lemma

H.C.F. of 625 and 3125 = 625

Now, consider 625 and 15625.

By applying Euclid’s division lemma

Therefore, H.C.F. of 625, 3125 and 15625 = 625

Hence, the required number is

#### Page No 1.22:

Find the greatest number that divides 445, 572 and 699 and leaves remainders of 4, 5 and 6 respectively.

The required number when divides 445, 572 and 699 leaves remainders 4, 5 and 6 this means  are completely divisible by the number.

Therefore, the required number = H.C.F. of 441, 567 and 693.

First consider 441 and 567.

By applying Euclid’s division lemma

Therefore, H.C.F. of 441 and 567 = 63

Now, consider 63 and 693

By applying Euclid’s division lemma

Therefore, H.C.F. of 441, 567 and 693 = 63

Hence, the required number is.

#### Page No 1.22:

Find the greatest number which divides 2011 and 2623 leaving remainder 9 and 5 respectively.

The required number when divides 2011 and 2623 leaves remainders 9 and 5 this means are completely divisible by the number.

Therefore, the required number = H.C.F. of 2002 and 2618

By applying Euclid’s division lemma

H.C.F. of 2002 and 2618 = 154

Hence, the required number is.

#### Page No 1.22:

We are given the length, breadth and height of a room as 8m 25cm, 6m 75cm and 4m 50cm, respectively. We need to determine the largest room which can measure the three dimensions of the room exactly.

We first convert each dimension in cm

Length of room = 8m 25cm = 825cm

Breadth of room = 6m 75cm = 675cm

Height of room = 4m 50cm = 450cm.

Therefore, the required longest rod = H.C.F. of 825, 675 and 450.

First we consider 675 and 450.

By applying Euclid’s division lemma

Therefore, H.C.F. of 675 and 450 = 225

Now, we consider 225 and 825.

By applying Euclid’s division lemma

Therefore, H.C.F. of 825, 675 and 450 = 75

Hence, the length of required longest rod is

#### Page No 1.22:

We are given that, 105 goats, 140 donkeys and 175 cows. There is only one boat which will have to make many y trips in order to do so. The lazy boatman has his own conditions for transporting them. He insists that he will take the same number of animals in every trip and they have to be of the same kind. He will naturally like to take the largest possible number each time. We need to tell the number of animals that went in each trip.

Given that

Number of goats = 105

Number of donkeys = 140

Number of cows = 175.

Therefore, the largest number of animals in 1 trip = H.C.F. of 105, 140 and 175.

First we consider 105 and 140.

By applying Euclid’s division lemma

Therefore, H.C.F. of 105 and 140 = 35

Now, we consider 35 and 175.

By applying Euclid’s division lemma

Therefore, H.C.F. of 105, 140 and 175 = 35

Hence, the number of animals went in each trip is.

#### Page No 1.22:

We are given that 15 pastries and 12 biscuit packets have been donated for a school fete. These are to be packed in several smaller identical boxes with the same number of pastries and biscuit packets in each. We need to find the number of biscuit packets and number of pastries each box contain.

Given that

Number of pastries = 15

Number of biscuits packets = 12.

Therefore, required number of boxes to contain equal number = H.C.F. of 15 and 12.

By applying Euclid’s division lemma

Therefore, number of boxes required = 3.

Hence each box will contain pastries and biscuits packets.

#### Page No 1.23:

A mason has to fit a bathroom with square marble tiles of the largest possible size. The size of the bathroom is 10ft. by 8ft. We need to find the size in inches of the tile required that has to be cut and number of such tiles are required.

Size of bathroom = 10ft by 8ft

The largest size of tile required = H.C.F. of 120 and 96.

By applying Euclid’s division lemma

Therefore, H.C.F. = 24.

Thus, largest size of tile required = 24 inches.

Therefore,

#### Page No 1.23:

We are given that two brands of chocolates are available in packs of 24 and 15 respectively. If he needs to buy an equal number of chocolates of both kinds, then find least number of boxes of each kind he would need to buy.

Given that

Number of chocolates of 1st brand in one pack =24

Number of chocolates of 2nd brand in one pack = 15.

Therefore, the least number of chocolates he need to purchase is

Therefore, the number of packet of 1st brand is

And the number of packet of 2nd brand is

.

#### Page No 1.23:

Given that 144 cartons of coke cans and 90 cartons of Pepsi cans are to be stacked in a canteen. If each stack is of the same height and contains cartons of the same drink We need to find the greatest number of cartons, each stack would have

Given that

Number of cartons of coke cans = 144

Number of cartons of Pepsi cans = 90.

Therefore, the greatest number of cartons in one stack = H.C.F. of 144 and 90.

By applying Euclid’s division lemma

Hence, the greatest number cartons in one stack

#### Page No 1.23:

We are given that during a sale, color pencils were being sold in packs of 24 each and crayons in packs of 32 each. If we want full packs of both and the same number of pencils and crayons, we need to find the number of each we need to buy.

Given that

Number of color pencils in one pack = 24

Number of crayons in pack = 32.

Therefore, the least number of both colors to be purchased

Hence, number of packs of pencils to bought

,

And number of packs of crayon to be bought

.

#### Page No 1.23:

The merchant has 3 different oils of 120 liters, 180 liters and 240 liters respectively.

So the greatest capacity of the tin for filling three different types of oil is given by the H.C.F. of 120,180 and 240.

So first we will calculate H.C.F of 120 and 180 by Euclid’s division lemma.

The divisor at the last step is 60. So the H.C.F of 120 and 180 is 60.

Now we will find the H.C.F. of 60 and 240,

The divisor at the last step is 60. So the H.C.F of 240 and 60 is 60.

Therefore, the tin should be of

#### Page No 1.29:

TO EXPRESS: each of the following numbers as a product of their prime factors

(i) 420

(ii) 468

(iii) 945

(iv) 7325

#### Page No 1.29:

TO EXPRESS: each of the following numbers as a product of their prime factors

(i) 20570

$20570=2×5×{11}^{2}×17$

(ii) 58500

(iii) 45470971

#### Page No 1.29:

EXPLAIN: Why and are composite numbers

We can see that both the numbers have common factor 7 and 1.

And we know that composite numbers are those numbers which have at least one more factor other than 1.

Hence after simplification we see that both numbers are even and therefore the given two numbers are composite numbers

#### Page No 1.29:

TO CHECK: Whether can end with the digit 0 for any natural number n.

We know that

Therefore, prime factorization of does not contain 5 and 2 as a factor together.

Hence can never end with the digit 0 for any natural number n

#### Page No 1.32:

TO FIND: LCM and HCF of following pairs of integers

TO VERIFY:

(i) 26 and 91

Let us first find the factors of 26 and 91

We know that,

Hence verified

(ii) 510 and 92

Let us first find the factors of 510 and 92

We know that,

Hence verified

(iii) 336 and 54

Let us first find the factors of 336 and 54

We know that,

Hence verified

#### Page No 1.32:

TO FIND: LCM and HCF of following pairs of integers

(i) 15, 12 and 21

Let us first find the factors of 15, 12 and 21

(ii) 17, 23 and 29

Let us first find the factors of 17, 23 and 29

(iii) 8, 9 and 25

Let us first find the factors of 8,9 and 25

(iv) 40, 36 and 126

Let us first find the factors of 40, 36 and 126

(v) 84, 90 and 120

Let us first find the factors of 84, 90 and 120

(vi) 24, 15 and 36

Let us first find the factors of 24, 15 and 36.

#### Page No 1.32:

TO FIND: Greatest number of 6 digits exactly divisible by 24, 15 and 36

The greatest 6 digit number be 999999

24, 15 and 36

Since

Therefore, the remainder is 279.

Hence the desired number is equal to

Hence is the greatest number of 6 digits exactly divisible by 24, 15 and 36.

#### Page No 1.32:

GIVEN: A rectangular yard is 18 m 72 cm long and 13 m 20 cm broad .It is to be paved with square tiles of the same size.

TO FIND: Least possible number of such tiles.

Length of the yard = 18 m 72 cm = 1800 cm  + 72 cm = 1872 cm               (∵ 1 m = 100 cm)

Breadth of the yard =13 m 20 cm = 1300 cm + 20 cm = 1320 cm

The size of the square tile of same size needed to the pave the rectangular yard is equal the HCF of the length and breadth of the rectangular yard.

Prime factorisation of 1872 =
Prime factorisation of 1320 =
HCF of 1872 and 1320 = ${2}^{3}×3=24$
∴ Length of side of the square tile = 24 cm

Number of tiles required =
Thus, the least possible number of tiles required is 4290.

#### Page No 1.33:

TO FIND: Least number that is divisible by all the numbers between 1 and 10 (both inclusive)

Let us first find the L.C.M of all the numbers between 1 and 10 (both inclusive)

1 = 1

2 = 2

3 = 3

4 = 22

5 = 5

6 = 2 × 3

7 = 7

8 = 23

9 = 32

10 = 2 × 5

Hence is the least number that is divisible by all the numbers between 1 and 10 (both inclusive)

#### Page No 1.33:

TO FIND: Smallest number that, when divided by 35, 56 and 91 leaves remainder of 7 in each case

L.C.M OF 35, 56 and 91

Hence 84 is the least number which exactly divides 28, 42 and 84 i.e. we will get a remainder of 0 in this case. But we need the smallest number that, when divided by 35, 56 and 91 leaves remainder of 7 in each case

Therefore

Hence is smallest number that, when divided by 35, 56 and 91 leaves remainder of 7 in each case.

#### Page No 1.33:

GIVEN: In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm and 90 cm.

TO FIND: minimum distance each should walk so that all can cover the same distance in complete steps.

The distance covered by each of them is required to be same as well as minimum. The required distance each should walk would be the L.C.M of the measures of their steps i.e. 80 cm, 85 cm, and 90 cm,

So we have to find the L.C.M of 80 cm, 85 cm, and 90 cm.

Hence minimum distance each should walk so that all can cove the same distance in complete steps

#### Page No 1.33:

TO FIND: The number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21.

L.C.M Of 8, 15 and 21.

When 110000 is divided by 840, the remainder is obtained as 800.

Now, 110000 − 800 = 109200 is divisible by each of 8, 15 and 21.

Also, 110000 + 40 = 110040 is divisible by each of 8, 15 and 21.

109200 and 110040 are greater than 100000.

Hence, 110040 is the number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21.

#### Page No 1.33:

TO FIND: The smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively.

L.C.M of 28 and 32.

Hence 224 is the least number which exactly divides 28 and 32 i.e. we will get a remainder of 0 in this case. But we need the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively

Therefore

Hence is the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively

#### Page No 1.33:

GIVEN: A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60, and 72 km a day, round the field.
TO FIND: When they meet again.

In order to calculate the time when they meet, we first find out the time taken by each cyclist in covering the distance.

Number of days 1st cyclist took to cover 360 km =

Similarly, number of days taken by 2nd cyclist to cover same distance =

Also, number of days taken by 3rd cyclist to cover this distance =
Now, LCM of

Thus, all of them will take 30 days to meet again.

#### Page No 1.33:

GIVEN: LCM and HCF of two numbers are 180 and 6 respectively. If one number is 30

TO FIND: Other number

We know that,

#### Page No 1.33:

GIVEN: HCF of two numbers is 16. If product of numbers is 3072

TO FIND: L.C.M of numbers

We know that,

#### Page No 1.33:

GIVEN: LCM and HCF of two numbers are 2175 and 145 respectively. If one number is 725

TO FIND: Other number

We know that,

#### Page No 1.33:

TO FIND: can two numbers have 16 as their H.C.F and 380 as their L.C.M

On dividing 380 by 16 we get 23 as the quotient and 12 as the remainder,

Since L.C.M is not exactly divisible by the H.C.F, two numbers cannot have 16 as their H.C.F and 380 as their L.C.M

#### Page No 1.33:

GIVEN: HCF of two numbers 306 and 657 is 9.

TO FIND: L.C.M of number

We know that,

#### Page No 1.33:

TO FIND: Smallest number which when increased by 17 is exactly divisible by both 520 and 468.

L.C.M OF 520 and 468

Hence 4680 is the least number which exactly divides 520 and 468 i.e. we will get a remainder of 0 in this case. But we need the Smallest number which when increased by 17 is exactly divided by 520 and 468.

Therefore

Hence is Smallest number which when increased by 17 is exactly divisible by both 520 and 468.

#### Page No 1.41:

Let us assume that is rational .Then, there exist positive co primes a and b such that

#### Page No 1.41:

(i) Let us assume that is rational .Then , there exist positive co primes a and b such that

(ii) Let us assume that is rational .Then , there exist positive co primes a and b such that

We know that is an irrational number

Here we see that is a rational number which is a contradiction

(iii) Let us assume that is rational. Then , there exist positive co primes a and b such that

Here we see that is a rational number which is a contradiction as we know that is an irrational number

(iv) Let us assume that is rational .Then, there exist positive co primes a and b such that

Here we see that is a rational number which is a contradiction as we know that is an irrational number

#### Page No 1.41:

Let us assume that is rational .Then, there exist positive co primes a and b such that

Here we see that is a rational number which is a contradiction as we know that is an irrational number.

#### Page No 1.41:

Let us assume that is rational. Then, there exist positive co primes a and b such that

Here we see that is a rational number which is a contradiction as we know that is an irrational number

#### Page No 1.41:

Let us assume that is rational .Then, there exist positive co primes a and b such that

Here we see that is a rational number which is a contradiction as we know that is an irrational number

#### Page No 1.42:

(i) Let us assume that is rational .Then , there exist positive co primes a and b such that

(ii) Let us assume that $\frac{3}{2\sqrt{5}}$  is rational .Then , there exist positive co primes a and b such that

(iii) Let us assume that is rational .Then , there exist positive co primes a and b such that

(iv) Let us assume that is rational .Then , there exist positive co primes a and b such that

#### Page No 1.42:

Let us assume that is rational .Then, there exist positive co primes a and b such that

This implies, $\sqrt{3}$ is a rational number, which is a contradiction.
Hence, is irrational number.

#### Page No 1.42:

Let us assume that is rational .Then, there exist positive co primes a and b such that

This implies,

#### Page No 1.42:

Let us assume that is rational .Then, there exist positive co primes a and b such that

#### Page No 1.42:

Let us assume that is rational .Then, there exist positive co primes a and b such that

#### Page No 1.42:

Let us assume that is rational .Then, there exist positive co primes a and b such that

#### Page No 1.42:

Let us assume that is rational .Then, there exist positive co primes a and b such that

#### Page No 1.48:

FUNDAMENTAL THEOREM OF ARITHMETIC:

Every composite number can be expressed (factorised) as a product of primes, and this factorization is unique except for the order in which the prime factors occur.

While writing a positive integer as the product of primes, if we decide to write the prime factors in ascending order and we combine the same primes, then the integer is expressed as the product of powers of primes and the representation is unique.

So,we can say that every composite number can be expressed as the products of powers distinct primes in ascending or descending order in a unique way.

#### Page No 1.48:

Using the factor tree for prime factorization, we have:

Therefore,

#### Page No 1.48:

Euclid’s Division Lemma:

Let a and b be any two positive integers.

Then, there exist unique integers q and r such that

,

If then .

Otherwise, r satisfies the stronger inequality.

#### Page No 1.48:

Using the factor tree for prime factorization, we have:

Therefore,

Hence the exponent of 2 in 144 is .

#### Page No 1.48:

Using the factor tree for prime factorization, we have:

Therefore,

The exponents of 2 and 7 are 1 and 2 respectively.

Hence the sum of the exponents is .

#### Page No 1.48:

Since, it is given that

Hence the number of consecutive zeroes are.

#### Page No 1.48:

It is given that the product of two numbers is 1080.

Let the two numbers be a and b.

Therefore,

HCF is 30.

We need to find the LCM

We know that the product of two numbers is equal to the product of the HCF and LCM.

Thus,

Hence the LCM is .

#### Page No 1.49:

We need to find the condition to be satisfied by q so that a rational number has a terminating decimal expression.

For the terminating decimal expression, we should have a multiple of 10 in the denominator.

Hence, the prime factorization of q must be of the form, where m and n are non-negative integers.

#### Page No 1.49:

We need to find the condition to be satisfied by q so that a rational number has a non-terminating decimal expression.

For the terminating decimal expression, we should not have a multiple of 10 in the denominator.

Hence, the prime factorization of q must not be of the form, where m and n are non-negative integers.

#### Page No 1.49:

We need to fill the values for a and b in the following factor tree:

It is clear from the factor tree above that

Also,

Thus, the missing entries are and.

#### Page No 1.49:

We have,

Theorem states:

Let be a rational number, such that the prime factorization of q is of the form , where m and n are non-negative integers.

Then, x has a decimal expression which terminates after k places of decimals, where k is the larger of m and n.

This is clear that the prime factorization of the denominator is of the form .

Hence, it has terminating decimal expansion which terminates after places of decimal.

#### Page No 1.49:

We have,

Theorem states:

Let be a rational number, such that the prime factorization of q is not of the form , where m and n are non-negative integers.

Then, x has a decimal expression which is non-terminating repeating.

This is clear that the prime factorization of the denominator is not of the form.

Hence, it has non-terminating decimal expansion.

#### Page No 1.49:

Let us simplify

is rational number

#### Page No 1.49:

Algorithm is a step-by-step procedure for calculations.

For example:

Euclid’s Division Algorithm: In order to compute the HCF of two positive integers say a and b, with by using Euclid’s algorithm, we follow the following steps:

STEP I : Apply Euclid’s Division Lemma to a and b and obtain whole numbers and ,such that ,

STEP II: If , b is the HCF of a and b.

STEP III: If , apply Euclid’s division lemma to b and and obtain whole numbers and ,such that

STEP IV: If , then is the HCF of a and b.

STEP V: If , then apply Euclid’s division lemma to and and continue the above process till the remainder is zero. The divisor at this stage i.e; , or the non-zero remainder at the previous stage is the HCF of a and b.

#### Page No 1.49:

A proven statement used as a stepping-stone toward the proof of another statement is called lemma.

For example:

Euclid’s Division Lemma: Let a and b be any two positive integers.

Then, there exist unique integers q and r such that

,

If then .

Otherwise, r satisfies the stronger inequality.

#### Page No 1.49:

It is given that p and q are two prime numbers; we have to find their HCF.

We know that the factors of any prime number are 1 and the prime number itself.

For example, let and

Thus, the factors are as follows

And

Now, the HCF of 2 and 3 is 1.

Thus the HCF of p and q is .

#### Page No 1.49:

It is given that p and q are two prime numbers; we have to find their LCM.

We know that the factors of any prime number are 1 and the prime number itself.

For example, let and

Thus, the factors are as follows

And

Now, the LCM of 2 and 3 is .

Thus the HCF of p and q is .

#### Page No 1.49:

We know that the factors of any prime number are 1 and the prime number itself.

For example, let

Thus, the factors are as follows

Hence, the total number of factors of a prime number is

#### Page No 1.49:

A composite number is a positive integer which has a divisor other than one or itself.

In other words a composite number is any positive integer greater than one that is not a prime number.

#### Page No 1.49:

The smallest composite number is 4

The smallest prime number is 2

Thus, the HCF of and is .

#### Page No 1.49:

HCF of two numbers is always a factor of their LCM

True

Reason:

The HCF is a factor of both the numbers which are factors of their LCM.
Thus the HCF is also a factor of the LCM of the two numbers.

#### Page No 1.49:

Hereis an irrational number

True

Reason:

Rational number is one that can be expressed as the fraction of two integers.

Rational numbers converted into decimal notation always repeat themselves somewhere in their digits.

For example, 3 is a rational number as it can be written as 3/1 and in decimal notation it is expressed with an infinite amount of zeros to the right of the decimal point. 1/7 is also a rational number. Its decimal notation is 0.142857142857…, a repetition of six digits.

However cannot be written as the fraction of two integers and is therefore irrational.

Now,

Thus, it is irrational.

#### Page No 1.49:

The sum of two prime numbers is always a prime number.

False

Reason:

Let us prove the above by taking an example.

Let the two given prime numbers be 2 and 7.

Thus, their sum, i.e; 9 is not a prime number.

Hence the above statement is false

#### Page No 1.49:

The product of any three natural numbers is divisible by 6.

True

Reason:

Let the three consecutive natural numbers be 1,2 and 3.

Their product is 6, which is divisible by 6

Let the other set of three consecutive natural numbers be 3, 4 and 5.

Their product is 60, which is divisible by 6

#### Page No 1.49:

Every even integer is of the form 2m, where m is an integer (True/False)

True

Reason:

Let the various values of m as -1, 0 and 9.

Thus, the values for 2m become -2, 0 and 18 respectively.

#### Page No 1.49:

Every odd integer is of the form, where m is an integer (True/False)

True

Reason:

Let the various values of m as -1, 0 and 9.

Thus, the values for become -3, -1 and 17 respectively.

These are odd integers.

#### Page No 1.49:

The product of two irrational numbers is an irrational number (True/False)

False

Reason:

Let us assume the two irrational numbers be and

Sometimes, it is and sometimes it isn't.

And are both irrational as their product is

Nowand are both irrational but their product, is rational (in fact, it equals 4)

#### Page No 1.49:

The sum of two irrational numbers is an irrational number (True/False)

False

Reason:
However,
is not rational because there is no fraction, no ratio of integers that will equal
. It calculates to be a decimal that never repeats and never ends. The same can be said for . Also, there is no way to write as a fraction. In fact, the representation is already in its simplest form.

To get two irrational numbers to add up to a rational number, you need to add irrational numbers such as and . In this case, the irrational portions just happen to cancel out leaving: which is a rational number (i.e. 2/1).

#### Page No 1.49:

We need to find the value of n, for which ends in 5.

Clearly,

Also, all the values of n will make end in 0.

Thus, there is no value of n for which ends in 5.

#### Page No 1.49:

It is given that a and b are two relatively prime numbers; we have to find their HCF.

We know that two numbers are relatively prime if they don’t have any common divisor.

Also, the factors of any prime number are 1 and the prime number itself.

For example, let a = 7 and b = 20

Thus, the factors are as follows
a = 7 × 1

And
b = 22 × 5 × 1

Now, the HCF of 7 and 20 is 1.

Thus the HCF of a and b is .

#### Page No 1.49:

It is given that a and b are two relatively prime numbers; we have to find their LCM.

We know that two numbers are relatively prime if they don’t have any common divisor.

Also, the factors of any prime number are 1 and the prime number itself.

For example, let a = 7 and b = 20

Thus, the factors are as follows
a = 7 × 1

And
b = 22 × 5 × 1

Now, the LCM of 7 and 20 is 140.

Thus the HCF of a and b is ab.

#### Page No 1.49:

Two numbers have 12 as their HCF and 350 as their LCM (True/False).

False.

Reason:

We know that HCF should divide LCM.

But, the HCF 12 does not divide the LCM 350.

#### Page No 1.50:

Using the factor tree for prime factorization, we have:

Therefore,

Thus, the exponent of 2 in 144 is 4.

Hence the correct choice is (a).

#### Page No 1.50:

It is given that the LCM of two numbers is 1200.

We know that the HCF of two numbers is always the factor of LCM

Checking all the options:

(a) 600 is the factor of 1200.

So this can be the HCF.

(b) 500 is not the factor of 1200.

So this cannot be the HCF.

(c) 400 is the factor of 1200.

So this can be the HCF.

(d) 200 is the factor of 1200.

So this can be the HCF.

Hence the correct choice is (b).

#### Page No 1.50:

Since, it is given that

n = ${2}^{3}×{3}^{4}×{5}^{4}×7$

So, this means the given number n will end with 3 consecutive zeroes.

#### Page No 1.50:

Using the factor tree for prime factorization, we have:

Therefore,

The exponents of 2 and 7 are 2 and 2 respectively.

Thus the sum of the exponents is

Hence the correct choice is (c).

#### Page No 1.50:

We have,

Theorem states:

Let be a rational number, such that the prime factorization of q is of the form, where m and n are non-negative integers.

Then, x has a decimal expression which terminates after k places of decimals, where k is the larger of m and n.

This is given that the prime factorization of the denominator is of the form.

Hence, it has terminating decimal expansion which terminates after places of decimal.

Hence, the correct choice is (b).

#### Page No 1.50:

Let the two odd prime numbers and be 5 and 3.

Then,

And

Thus,

16 is even number.

Take another example, with and be 11 and 7.

Then,

And

Thus,

72 is even number.

Thus, we can say that is even number

In general the square of odd prime number is odd. Hence the difference of square of two prime numbers is odd

Hence the correct choice is (a).

#### Page No 1.50:

Two positive integers are expressed as follows:

p and q are prime numbers.

Then, taking the highest powers of p and q in the values for a and b we get:

LCM

Hence the correct choice is (c).

#### Page No 1.50:

Two positive integers are expressed as follows:

p and q are prime numbers.

Then, taking the smallest powers of p and q in the values for a and b we get

HCF

Hence the correct choice is (a).

#### Page No 1.50:

Two positive integers are expressed as follows:

p and q are prime numbers.

Then, taking the smallest powers of p and q in the values for m and n we get

HCF

Hence the correct choice is (b).

#### Page No 1.50:

LCM

HCF

We know that the product of numbers is equal to the product of their HCF and LCM.

Therefore,

Hence the correct choice is (c).

#### Page No 1.51:

Using the factor tree for 95, we have:

Using the factor tree for 152, we have:

Therefore,

HCF

Hence the correct choice is (c).

#### Page No 1.51:

HCF

We have to find the value for LCM

We know that the product of numbers is equal to the product of their HCF and LCM.

Therefore,

Hence the correct choice is (c).

#### Page No 1.51:

LCM   ……(I)

We have to find the value for n

Also

We know that the while evaluating LCM, we take greater exponent of the prime numbers in the factorization of the number.

Therefore, by applying this rule and taking we get the LCM as

LCM ……(II)

On comparing (I) and (II) sides, we get:

Hence the correct choice is (b).

#### Page No 1.51:

We have,

Theorem states:

Let be a rational number, such that the prime factorization of q is of the form, where m and n are non-negative integers.

Then, x has a decimal expression which terminates after k places of decimals, where k is the larger of m and n.

This is given that the prime factorization of the denominator is of the form.

Hence, it has terminating decimal expansion which terminates after places of decimal.

Hence, the correct choice is (d).

#### Page No 1.51:

We know that the co-prime numbers have no factor in common, or, their HCF is 1.

Thus,and have the same factors with twice of the exponents of p and q respectively, which again will not have any common factor.

Thus we can conclude that and are co-prime numbers.

Hence, the correct choice is (a).

#### Page No 1.51:

(i) We have,

Theorem states:

Let be a rational number, such that the prime factorization of q is not of the form, where m and n are non-negative integers.

Then, x has a decimal expression which does not have terminating decimal.

(ii) We have,

Theorem states:

Let be a rational number, such that the prime factorization of q is not of the form, where m and n are non-negative integers.

Then, x has a decimal expression which does not have terminating decimal.

(iii) We have,

Theorem states:

Let be a rational number, such that the prime factorization of q is not of the form, where m and n are non-negative integers.

Then, x has a decimal expression which does not have terminating decimal.

(iv) We have,

Theorem states:

Let be a rational number, such that the prime factorization of q is of the form, where m and n are non-negative integers.

Then, x has a decimal expression which terminates after k places of decimals, where k is the larger of m and n.

Then, x has a decimal expression which will have terminating decimal after 3 places of decimal.

Hence the (iv) option will have terminating decimal expansion.

There is no correct option.

#### Page No 1.51:

Since

The least prime factor of has to be 2; unless is a prime number greater than 2.

Suppose is a prime number greater than 2. Then must be an odd number

o one of a or b must be an even number.

Suppose then that a is even. Then the least prime factor of a is 2; which is not 3 or 7. So a can not be an even number nor can b be an even number. Hence can not be a prime number greater than 2 if the least prime factor of a is 3 and b is 7.

Hence the correct choice is (a).

#### Page No 1.51:

We have,

Let

Then,

Subtract these to get

Thus, we can also conclude that all infinite repeating decimals are rational numbers.

Hence, the correct choice is (b).

#### Page No 1.51:

Out of the given choices is the only smallest number by which if we multiply we get a rational number.

Hence, the correct choice is (c).

#### Page No 1.51:

For terminating the decimal expansion after one place of decimal, the highest power of m and n in should be 1.

Let

We will get:

Thus, it is evident that we multiplied it by

Hence, the correct choice is (a).

#### Page No 1.51:

We know that is always divisible by both and .

So, is always divisible by both and.

Hence, the correct choice is (c).

#### Page No 1.51:

We know that will end in 6

And will end in 5.

Now, always end with

Hence the correct choice in (a).

#### Page No 1.52:

Let the two numbers be a and b.

(a) If we assume that the a and b are prime.

Then,

HCF

LCM

(b) If we assume that a and b are co-prime.

Then,

HCF

LCM

(c) If we assume that a and b are composite.

Then,

HCFor any other highest common integer

LCM

(d) If we assume that a and b are equal and consider a=b=k.

Then,

HCF

LCM

Hence the correct choice is (d).

#### Page No 1.52:

Let the HCF be x and the LCM of the two numbers be y.

It is given that the sum of the HCF and LCM is 1260

…… (i)

And, LCM is 900 more than HCF.

…… (ii)

Substituting (ii) in (i), we get:

Substituting in (ii), we get:

We also know that the product the two numbers is equal to the product of their LCM and HCF

Thus the product of the numbers

Hence, the correct choice is (b).

#### Page No 1.52:

Any prime number greater than 3 is of the form, where k is a natural number.

Thus,

When, is divided by 6, we get, and remainder as 1.

Hence, the correct choice is (a).

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