Question 677483
If x was 504 then the product of 504 and x would be {{{504*504 = 504^2}}} which is clearly a perfect square. But is 504 the smallest possible value for x that works?<br>
To answer this we will take advantage of the fact that <u>perfect squares can be expressed as the product of perfect squares</u>. 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.