Prime factorize the 450:

450 = 50*9 = 2 * 3^2 * 5^2

To get a perfect cube, we need to multiply by a number that will make every exponent in the prime factorization into some multiple of 3. So the smallest thing we can multiply by is 2^2 * 3^1 * 5^1 = 60, since that will give us 2^3 * 3^3 * 5^3, which is the cube of (2)(3)(5).

swerve wrote:What is the smallest positive integer \(x\) such that \(450x\) is the cube of a positive integer?

A. 2
B. 15
C. 30
D. 60
E. 120

The OA is D

Source: Manhattan Prep

We must remember that all perfect cubes break down to unique prime factors, each of which has an exponent that is a multiple of 3. So, let's break down 450 into primes to help determine what extra prime factors we need to make 450x a perfect cube.

450 = 45 x 10 = 9 x 5 x 5 x 2 = 3 x 3 x 5 x 5 x 2 = 5^2 x 3^2 x 2^1

In order to make 450x a perfect cube, we need two more 2s, one more 3, and one more 5. Thus, the smallest perfect cube that is a multiple of 450 is 5^3 x 3^3 x 2^3. In other words, 450x = (5^3)(3^3)(2^3). Thus:

x = (5^3 * 3^3 * 2^3)/450

x = (5^3 * 3^3 * 2^3)/(5^2 * 3^2 * 2^1)

x = 5^1 * 3^1* 2^2 = 60

Answer: D