the smallest whole number that is to be multiplied with 59535 to make a perfect square number is x .the sum of digits of x is

Question

Answer is 6 ..please give the solution step by step so that it can help me understand clearly 

Answer 1

59535 = 5(11907) = (5)(3)(3969)

 

                           = (5)(3)(3)(1323)

 

                           = (5)(3)(3)(3)(441)

 

                           = (5)(3)(3)(3)(3)(147)

 

                           = (5)(3)(3)(3)(3)(7)(3)(7)

 

                           = 7²3⁵5

 

We will have a perfect square if all of the exponents are even.  so, multiplying by 15 = (3)(5) gives 7²3⁶5² = [7(3)³(5)]²

 

The sum of the digits of 15 is 6.

 

Answer 2

When expressed as the product of its prime factors, the number 59535 = 3*3*3*3*3*5*7*7.

 

Now, to make a perfect square, the factors must appear in sets of two [that is, an even number of times]:

 

    (3*3*3*3*3*5*7*7)*(3*5) = (3*3*3*5*7)*(3*3*3*5*7)

                                                a perfect square

 

The smallest integer to be multiplied by 59535 to make a perfect square is (3*5)=15.  The sum of the digits of 15 is 6.