Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
The brute-force approach would be to systematically list multiples of 450 from 450 on up, test each one to see whether it is a perfect cube (the cube of a positive integer), and choose the first multiple that meets the criterion. However, this approach is very cumbersome. Even just trying the answer choices would take a long time. In fact, without insight into the nature of cubes, it is difficult to see how we can easily test whether a number is a cube, except by cubing various integers and comparing the results to the number in question.
A more efficient approach takes advantage of a key property of perfect cubes: its prime factors come in triplets. In other words, each of its prime factors occurs 3 times (or 6 times, 9 times, etc.) in the cube’s prime factorization. To see why, try cubing 6 = (2•3):
6•6•6 = (2•3)(2•3)(2•3) = (2•2•2)(3•3•3).
As you can see, the 2’s and 3’s occur in triplets. So our goal is to make the prime factors of 450x occur in triplets as well.
The first step is to break up 450 into its prime factors:
450 = (45)(10) = (3•3•5)(2•5) = 2•3•3•5•5.
How many of each prime factor do we need to complete all the triplets? We are evidently missing two 2’s, one 3, and one 5. Multiplying these missing factors together, we get
2•2•3•5 = 60.
The correct answer is D.