Problem Solving

VJesus12 wrote:If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be

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

[spoiler]OA=E[/spoiler]

Source: Official Guide

This problem is testing us on the rule that when we express a perfect square by its unique prime factors, every prime factor's exponent is an even number.

Let's start by prime factorizing 3,150.

3,150 = 315 x 10 = 5 x 63 x 10 = 5 x 7 x 3 x 3 x 5 x 2

3,150 = 2^1 x 3^2 x 5^2 x 7^1

Notice that the exponents of both 2 and 7 are not even numbers. This tells us that 3,150 itself is not a perfect square. However, we are given that the product of 3,150 and y is a perfect square. Thus, we can write this as:

2^1 x 3^2 x 5^2 x 7^1 x y = square of an integer

According to our rule, we need all unique prime factors' exponents to be even numbers. Remember also that y is the smallest positive integer that makes 3150y a perfect square. Thus, we need one more 2 and one more 7. Therefore, y = 7 x 2 = 14

Answer: E