If x is a positive integer, for how many different values of x is square root 48/x a whole number?

1. We want all values of x such that  is a whole number.

2. Fist of all, x must be a factor of 48 

48=2*24=2*8*3=2*(2*2*2)*3=. This is called the prime factorization, so we write 48 as a multiplication of its prime factors so that we can write all the factors in an organized form without missing any.

All factors of 48 with only 2's are     {2, 4, 8, 16}
All factors of 48 with 2's and 3 are  {6, 12, 24, 48}
And there is 1 and 3.

3. We will check all of these factors as 48/x and see whether the division is a perfect square, so that the square of it is a hole number.

4.

48/2=24
48/4=12
48/8=6
48/16=3
48/6=8
48/12=4  a perfect square
48/24=2
48/48=1  a perfect square
48/3=16  a perfect square

so for x=12, x= 48 or x=3 , is a whole number. 

3 values

Here are the answers to the ones that i could see.