Problem 18 · 2018 Math Kangaroo
Hard
Number Theory
factorizationprimes
Which of the following numbers is not a factor of \(18^{2017} + 18^{2018}\)?
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Answer: C — 28
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Hint 1 of 2
Factor out the common power of 18 first.
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Hint 2 of 2
18 = 2·3², so see which prime each option needs.
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Approach: factor the sum and check prime content of each option
- 18^2017 + 18^2018 = 18^2017(1+18) = 18^2017 · 19 = 2^2017 · 3^4034 · 19.
- Its prime factors are only 2, 3 and 19.
- Among the options, 28 = 2²·7 needs a factor of 7, which is absent.
- So 28 is not a factor.
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