Q Find the greatest positive integer which divides 332 and 447 leaving remainder 2 in each case - Maths - Real Numbers

Question

Q Find the greatest positive integer which divides 332 and 447
leaving remainder
2 in each case
Q Find the largest number that divides 1251 , 9377 and 15628
leaving remainder
1,2 and 3 respectively
Q There are 48 bottles of products A and 108 bottles of
products B These are
to be packed in identical boxes with same number of bottle ,
one particular
product being in one box Find the maximum number of bottles
that should
be kept in one pack so that no bottle of product A or B is
left behind
Q There are 24 nuts of same size and 32 bolts of same size It
is required to
packet contains equal number of nuts or bolts What is the
maximum number of
nuts or bolts that should be kept in each packet?Also find the
minimum number
of packets that should be formed
Q A sweet seller has 420 kaju barfis and 130 badam barfis She
wants to stack
them in such a way that each stack has same number and they
take up the least
area of the tray What is the number of that can be in each
stack for this
purpose?
Q Prove that the square of any positive integer is of the form
5q,5q+1,5q+4 for
some integer q

Priyanka Kedia · Accepted Answer

$(2) S i n c e t h e n u m b e r t h a t d i v i d e s 1251, 9377 a n d 15628 l e a v i n g r e m a i n d e r 1, 2 a n d 3 r e s p e c t i v e l y T h u s, t h e r e q u i r e d n u m b e r s h o u l d d i v i d e (1251 - 1), (9377 - 2), a n d (15628 - 3) e x a c t l y S o, Re q u i r e d n u m b e r = H . C . F . o f 1250, 9375, a n d 15625 1250 = 5^{4} \times 2 9375 = 5^{4} \times 3 15625 = 5^{6} \Rightarrow H . C . F . o f 1250, 9375, a n d 15625 = 5^{4} = 625 H e n c e, t h e l \arg e s t n u m b e r t h a t d i v i d e s 1251, 9377 a n d 15628 l e a v i n g r e m a i n d e r 1, 2 a n d 3 r e s p e c t i v e l y i s 625$

$(5) A r e a o f t r a y w i l l b e l e a s t i f s w e e t s t a c k h a v e t h e s a m e n u m b e r o f b a r f i e s . S o, t h e n u m b e r o f s t a c k s w i l l b e l e a s t i f n u m b e r o f b a r f i e s i n e a c h s t a c k i s e q u a l t o t h e h c f o f 420 a n d 130. T h u s, H . C . F . o f 420 a n d 130 i s, 420 = 2^{2} \times 3 \times 5 \times 7 130 = 2 \times 5 \times 13 S o, H . C . F . o f 420 a n d 130 = 2 \times 5 = 10 H e n c e, e a c h s t a c k m u s t h a v e 10 b a r f i e s .$

$(6) L e t a b e a n y o d d p o s i t i v e int e g e r a n d b = 5 S o, a p p l y i n g d i v i s i o n l e m m a t o a a n d b, w e h a v e, a = 5 k + r, w h e r e m i s s o m e int e g e r a n d 0 \leq r < 5 T h u s, C a s e (1) . W h e n r = 0 a = 5 k + 0 \Rightarrow a = 5 k \Rightarrow a^{2} = {(5 k)}^{2} = 25 k^{2} = 5 (5 k^{2}) = 5 q (l e t 5 k^{2} = q, q \in N) C a s e (2) . W h e n r = 1 a = 5 k + 1 \Rightarrow a^{2} = {(5 k + 1)}^{2} = 25 k^{2} + 1 + 10 k = 5 k (5 k + 2) + 1 = 5 q + 1 (l e t k (5 k + 2) = q, q \in N) C a s e (3) . W h e n r = 2 a = 5 k + 2 \Rightarrow a^{2} = {(5 k + 2)}^{2} = 25 k^{2} + 4 + 20 k = 5 k (5 k + 4) + 4 = 5 q + 4 (l e t k (5 k + 4) = q, q \in N) C a s e (4) . W h e n r = 3 a = 5 k + 3 \Rightarrow a^{2} = {(5 k + 3)}^{2} = 25 k^{2} + 9 + 30 k = 5 (5 k^{2} + 6 k + 1) + 4 = 5 q + 4 (l e t 5 k^{2} + 6 k + 1 = q, q \in N) C a s e (4) . W h e n r = 4 a = 5 k + 4 \Rightarrow a^{2} = {(5 k + 4)}^{2} = 25 k^{2} + 16 + 40 k = 5 (5 k^{2} + 8 k + 3) + 1 = 5 q + 1 (l e t 5 k^{2} + 8 k + 3 = q, q \in N) H e n c e, s q u a r e o f a n y odd int e g e r i s i n t h e f o r m o f 5 q, 5 q + 1 a n d 5 q + 4$