Difference between revisions of "2007 AMC 10A Problems/Problem 17"

Revision as of 13:41, 3 June 2021

Problem

Suppose that $m$ and $n$ are positive integers such that $75m = n^{3}$ . What is the minimum possible value of $m + n$ ?

$\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 60 \qquad \text{(E)}\ 5700$

Solution

$3 \cdot 5^2m$ must be a perfect cube, so each power of a prime in the factorization for $3 \cdot 5^2m$ must be divisible by $3$ . Thus the minimum value of $m$ is $3^2 \cdot 5 = 45$ , which makes $n = \sqrt[3]{3^3 \cdot 5^3} = 15$ . The minimum possible value for the sum of $m$ and $n$ is $60\ \mathrm{(D)}$ .

2007 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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All AMC 10 Problems and Solutions

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

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 == Problem ==
-Suppose that <math>m</math> and <math>n</math> are positive [[integer]]s such that <math>75m = n^{3}</math>. What is a minimum possible value of <math>m + n</math>?
+Suppose that <math>m</math> and <math>n</math> are positive [[integer]]s such that <math>75m = n^{3}</math>. What is the minimum possible value of <math>m + n</math>?
 <math>\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 60 \qquad \text{(E)}\ 5700</math>

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Difference between revisions of "2007 AMC 10A Problems/Problem 17"

Revision as of 13:41, 3 June 2021

Problem

Solution

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