Problem 24 · 2012 Math Kangaroo
Stretch
Number Theory
last-digitfactorization
What is the last non zero digit of \(K = 2^{59} \times 3^{4} \times 5^{53}\)?
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Answer: C — 4
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Hint 1 of 2
Pair up 2s and 5s to make 10s, which only add trailing zeros.
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Hint 2 of 2
What's left after pulling out the 10s decides the last non-zero digit.
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Approach: strip out factors of 10, then read the last digit
- Write K = 2^59 · 3^4 · 5^53 = (2^53 · 5^53) · 2^6 · 3^4 = 10^53 · (2^6 · 3^4).
- The 10^53 only contributes trailing zeros, so the last non-zero digit comes from 2^6 · 3^4 = 64 · 81 = 5184.
- Its last digit is 4 (C).
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