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

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