1985 AHSME Problems/Problem 12

Problem

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?

$\mathrm{(A)\ } p^8q^8r^8 \qquad \mathrm{(B) \ }(pq^2r^2)^3 \qquad \mathrm{(C) \ } (p^2q^2r^2)^3 \qquad \mathrm{(D) \ } (pqr^2)^3 \qquad \mathrm{(E) \ }4p^3q^3r^3$

Solution

For a number of the form $p^aq^br^c$ to be a perfect cube and a multiple of $pq^2r^4$ , $a, b,$ and $c$ must all be multiples of $3$ , $a\ge1$ , $b\ge2$ , and $c\ge4$ . The smallest multiple of $3$ greater than $1$ is $3$ , the smallest multiple of $3$ greater than $2$ is $3$ , and the smallest multiple of $3$ greater than $4$ is $6$ . Therefore, the smallest such $p^aq^br^c$ is $p^3q^3r^6=(pqr^2)^3, \boxed{\text{D}}$ .

1985 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

1985 AHSME Problems/Problem 12

Problem

Solution

See Also