The least number of numbers to be deleted from the set (1,2,3,4,.......,14,15) so that the product of the remaining numbers is a perfect square,is
Answer :
We have set = ( 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 )
And wants to delete minimum number from this set as we get product of remaining numbers is a perfect square .
First we make prime factors of each number from 1 to 15 As :
1 = 1
2 = 1 2
3 = 1 3
4 = 1 2 2
5 = 1 5
6 = 1 2 3
7 = 1 7
8 = 1 2 2 2
9 = 1 3 3
10 = 1 2 5
11 = 1 11
12 = 1 2 2 3
13 = 1 13
14 = 1 2 7
15 = 1 3 5
Here we can have minimum 2 same kind of numbers to get multiplication is a perfect square : So
We can't have 11 and 13
Now we have : ( 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 12 , 14 , 15 )
Here we have total number of 2 = 11 , So we can keep maximum 10 , As we get 2 ( 5 ) = 10
And
Here we have total number of 3 = 6 , So we can keep all 3 , As we get 2 ( 3 ) = 6 .
And
Here we have total number of 5 = 3 , So we can keep maximum 2 , As we get 2 ( 1 ) = 2
And
Here we have total number of 7 = 2 , So we can keep all 7 , As we get 2 ( 1 ) = 2
As we want to delete minimum number and we also want to remove one of the 2 and one of the 5 , And In 10 we get ( 2 5 )
SO, we neglect 10 also , than we get
( 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 12 , 14 , 15 )
After multiplication , we get
1 2 3 4 5 6 7 8 9 12 14 15 = 914457600 = 302402
We have set = ( 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 )
And wants to delete minimum number from this set as we get product of remaining numbers is a perfect square .
First we make prime factors of each number from 1 to 15 As :
1 = 1
2 = 1 2
3 = 1 3
4 = 1 2 2
5 = 1 5
6 = 1 2 3
7 = 1 7
8 = 1 2 2 2
9 = 1 3 3
10 = 1 2 5
11 = 1 11
12 = 1 2 2 3
13 = 1 13
14 = 1 2 7
15 = 1 3 5
Here we can have minimum 2 same kind of numbers to get multiplication is a perfect square : So
We can't have 11 and 13
Now we have : ( 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 12 , 14 , 15 )
Here we have total number of 2 = 11 , So we can keep maximum 10 , As we get 2 ( 5 ) = 10
And
Here we have total number of 3 = 6 , So we can keep all 3 , As we get 2 ( 3 ) = 6 .
And
Here we have total number of 5 = 3 , So we can keep maximum 2 , As we get 2 ( 1 ) = 2
And
Here we have total number of 7 = 2 , So we can keep all 7 , As we get 2 ( 1 ) = 2
As we want to delete minimum number and we also want to remove one of the 2 and one of the 5 , And In 10 we get ( 2 5 )
SO, we neglect 10 also , than we get
( 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 12 , 14 , 15 )
After multiplication , we get
1 2 3 4 5 6 7 8 9 12 14 15 = 914457600 = 302402