The Quest for 700: Weekly GMAT Challenge (Answer)

Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:

The first task is to figure out a positive integer n for which the given condition is true: that all the unique positive divisors multiply up to n2.
Let’s first take a number-testing approach.

If n = 1, the condition holds, but we can quickly see that all the answer choices are equal in this case.
If n = 2, the condition doesn’t hold. Take the unique positive factors: 1 and 2. The product is 2, not 22.
The same is true for n = 3 or for any other prime number.
If n = 4, the condition still doesn’t hold. The unique positive divisors are 1, 2, and 4, which multiply up to 8, not 16.
If n = 6, the condition holds (thankfully). The unique positive divisors are 1, 2, 3, and 6, which multiply up to 36, or 62.

Thus, we can solve the problem if we figure out the product of all the unique positive divisors of 36. We don’t need the digits; we just need to know what that number is as a power of 6 (our n). The fast, organized way is to take the factors in pairs:

1 × 36 = 36 (=62)
2 × 18 = 36 (=62)
3 × 12 = 36 (=62)
4 × 9 = 36 (=62)

And finally there’s a 6, of course, the square root of 36. We only count this once (the question specifies unique divisors).
The product of the four “6-squared’s” (62) and the final 6 is 62 × 62 × 62 × 62 × 6 = 69.
This gives us the correct answer of E.

Alternatively, we could solve this problem in a more abstract way. First, we can figure out that n must be the product of two distinct primes, say p and q (so that n = pq). The only unique positive factors of n would then be 1, p, q, and pq (that is, n itself). Then we could write n2 as p2q2. (By the way, if n were allowed to be a cube of a prime number (such that n = p3), then the product of all of n‘s factors would also be n2, but this possibility is explicitly ruled out.)
In terms of p and q, then, the unique factors of n2 would then be as follows:

1, p, p2
q, qp, qp2
q2, q2p, q2p2
The product of all these factors is q9p9, or n9.

The correct answer is E.

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