Problem Solving

If y is the smallest positive integer such that 3150 mult by y is the square of an integer, then y must be:

A) 2
B) 5
C) 6
D) 7
E) 14

I understand that i will need to break down the prime factorization of 3150, but am stuck after that. don't know what to do...
thanks so much in advance.

3150=2*5*5*3*3*7
to be perfact square we need to mutiply this no by 2*7=14

we need to find out the prime factors of 3150

-> 3150 = 315 * 10 = 63 * 5 * 2 * 5= 3 * 21 * 5 * 2 * 5 = 3*3*7*5*2*5 = 2 * 3^2 * 5^2 * 7

for this to be a perfect square, we need to make 2 and 7 power of 2, which can be done by multiplying the expressin by 2*7

-> 14

so why can't i use the other prime factors? why only 2 and 7? thanks so much.

mstone wrote:so why can't i use the other prime factors? why only 2 and 7? thanks so much.

the question asks us what number multiplied by 3150 will give us a square number i.e. a result of an integer raised to the power 2

for this two happen, all the prime factors of 3150*y have to be raised to the power 2 or 4 or ....

since can be broken down to (2 * 3^2 * 5^2 * 7), we see that only 2 and 7 are not raised to the second power.

Since we have to find the smallest number ,y, which when multiplied with 3150 will give us a square of an integer, the answer is 2 * 7, which will give a square.

Hope this helped u understand why we use 2 and 7