Problem 16 · 2023 Math Kangaroo
Medium
Number Theory
number-systemsfactorization
The number \(5^{5^6}\) is to be written in the form \(n^n\) where n is a natural number. What is the value of n?
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Answer: A — \(5 \cdot 5^4\)
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Hint 1 of 2
Write n as a power of 5 and compare exponents.
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Hint 2 of 2
If n = 5k, then nn = 5k·5^k; you need k·5k = 56.
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Approach: match exponents by writing n as a power of 5
- Let n = 5k; then nn = 5k·5^k, and we need k·5k = 56.
- Taking k = 5 gives 5·55 = 56, so n = 55 = 5·54.
- So n = 5·54.
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