Difference between revisions of "1985 AHSME Problems/Problem 12"
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==Problem== | ==Problem== | ||
| − | Let | + | Let <math> p, q </math> and <math> r </math> be distinct [[prime number]]s, where <math> 1 </math> is not considered a prime. Which of the following is the smallest positive [[perfect cube]] having <math> n=pq^2r^4 </math> as a [[divisor]]? |
<math> \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 </math> | <math> \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 </math> | ||
Revision as of 23:54, 2 April 2018
Problem
Let 
and 
be distinct prime numbers, where 
is not considered a prime. Which of the following is the smallest positive perfect cube having 
as a divisor?
Solution
For a number of the form 
to be a perfect cube and a multiple of 
, 
and 
must all be multiples of 
, 
, 
, and 
. The smallest multiple of greater than is , the smallest multiple of greater than 
is , and the smallest multiple of greater than 
is 
. Therefore, the smallest such is 
.
See Also
| 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. 
