1985 AMC 12 Problem 12

Attempt Problem 12 of the 1985 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1985 AMC 12 solutions, or check the answer key.

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12.

Let

p,

q

and

r

be distinct prime numbers, where

1

is not considered a prime. Which of the following is the smallest positive perfect cube having

n=pq^2r^4

as a divisor?

p^8q^8r^8

(pq^2r^2)^3

(p^2q^2r^2)^3

(pqr^2)^3

4p^3q^3r^3

Every prime exponent in a perfect cube is a multiple of

3

Raise the exponents

1,

2,

and

4

to the least larger multiples of

3

A perfect cube has prime exponents divisible by

3.

The least multiples of

3

that are at least

1,

2,

and

4

are

3,

3,

and

6,

respectively. Thus the smallest possible cube is

p^3q^3r^6=(pqr^2)^3.

Therefore the correct answer is D.

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