SOLUTION: The product of 504 and a anumber X is a square number. What is the smallest possible value of X?

Question 677483: The product of 504 and a anumber X is a square number. What is the smallest possible value of X?
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
If x was 504 then the product of 504 and x would be which is clearly a perfect square. But is 504 the smallest possible value for x that works?

To answer this we will take advantage of the fact that perfect squares can be expressed as the product of perfect squares. So let's see how many perfect square factors there are in 504. 4 is a perfect square and, if you know your divisibility rules, we can see that 4 is a factor of 504:
504 = 4 * 126
4 is not a factor of 126 so it is a factor of 504 just once. 9 is a perfect square and, if you know your divisibility rules, we can see that 9 is a factor of 126:
504 = 4 * 9 * 14
There are no perfect factors of 14 (other than 1). So we have found all the perfect square factors of 504. The value for x we are looking for will turn the 14 into a perfect square. In other words x = 14.
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