Problem 18 · 2023 Math Kangaroo
Hard
Number Theory
last-digit
We call a positive integer powerfree if none of its digits can be written as a power of an integer with an exponent bigger than 1. For example, the number 53 is powerfree, but the number 54 is not powerfree since \(4 = 2^2\). Which one of the following numbers is the difference between the biggest and the smallest two-digit powerfree numbers?
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Answer: B — 55
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Hint 1 of 2
A digit is “bad” if it is a perfect power: 0, 1, 4, 8, 9.
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Hint 2 of 2
Find the largest and smallest two-digit numbers whose digits are all allowed, then subtract.
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Approach: restrict to allowed digits
- Digits that are powers (exponent>1) are 0=0², 1=1², 4=2², 8=2³, 9=3².
- Allowed digits are 2, 3, 5, 6, 7.
- The biggest two-digit powerfree number is 77 and the smallest is 22.
- Their difference is 77 − 22 = 55.
Mark:
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