Find the smallest positive integer

Question

find the smallest positive integer,N, such that the product of 135 and N is a perfect square.

Answer 1

A perfect square is a number that can be expressed as the product of two equal integers.

 

The product 135N is a perfect square.

 

The factors of 135 are groups of:   1*3*3*3*5

 

So, the smallest two equal integers that multiply to 135N are:

    1*3*3*5     and   1*3*3*5      [two 1's, four 3's, two 5's; must be even to split]

       1*3*3*5 * 1*3*3*5    = 1*3*3*3*5*N = 135N

          N = 1*3*5

          N = 15

Answer 2

Pick any integer value in the interval   N>0.  Multiply it by 135.  Then take the square-root of the result.  If the square-root of the result is an integer, then you found your smallest value of N.

 

Start with N=2, since 135 is not a perfect square.