If d is the smallest positive integer such that 3,150 multiplied by d is the square of an integer, then d must be
Choose the correct choice from the following options:
A) 2	B) 5	C) 6	D) 7	E) 14
Solution:   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.)   We also are given that 3,150 multiplied by d is the square of an integer. We can write this as:   2^1 x 3^2 x 5^2 x 7^1 x d = square of an integer   According to our rule, we need all unique prime factors' exponents to be even numbers. Thus, we need one more 2 and one more 7. Therefore, d = 7 x 2 = 14   Answer is 14.
<result>E</result>
If d is the smallest positive integer such that 3,150 multiplied by d is the square of an integer, then d must be