Skip to main content

2003 AMC 10B Problem 14

Problem 14 of 25IntermediateAlgebraNumber Theory

Given that 38⋅52=ab,3^8 \cdot 5^2 = a^b, where both aa and bb are positive integers, find the smallest possible value for a+b.a+b.

Answer choices

Show solution

Solution

Because aa must be divisible by 5,5, and 38⋅523^8 \cdot 5^2 is divisible by 525^2 but not 53,5^3, we need b≤2.b\le 2. Taking b=2b=2 gives a=38⋅52=34⋅5=405,a=\sqrt{3^8 \cdot 5^2}=3^4 \cdot 5=405, so a+b=407.a+b=407. This beats b=1,b=1, which gives a+b=164,026.a+b=164{,}026. Thus, the correct answer is D.

More practice

Concepts: prime factorization · perfect power · optimization

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.