2003 AMC 10B Problem 14 — Solution | LIVE by Po-Shen Loh

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2003 AMC 10B Problem 14

Below is the professionally curated solution for Problem 14 of the 2003 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 10B solutions, or check the answer key.

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Concepts:prime factorization perfect power optimization

Difficulty rating: 1390

14.

Given that $3^8 \cdot 5^2 = a^b,$ where both $a$ and $b$ are positive integers, find the smallest possible value for $a+b.$

$25$

$34$

$351$

$407$

$900$

Solution:

Because $a$ must be divisible by $5,$ and $3^8 \cdot 5^2$ is divisible by $5^2$ but not $5^3,$ we need $b\le 2.$

Taking $b=2$ gives $a=\sqrt{3^8 \cdot 5^2}=3^4 \cdot 5=405,$ so $a+b=407.$ This beats $b=1,$ which gives $a+b=164{,}026.$

Thus, the correct answer is D.

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