Problem 23 · 2012 Math Kangaroo
Stretch
Number Theory
factorization
Peter wrote the number 2012 in the form \(2012 = m^{m}(m^{k} - k)\) where m and k are natural numbers. Find the value of k.
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Answer: D — 9
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Hint 1 of 2
Factor 2012 and try to write it as m^m times (m^k − k).
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Hint 2 of 2
The m^m factor must divide 2012; test small m.
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Approach: factor 2012 and match the form
- Factor 2012 = 4 × 503 = 2² × 503.
- Take m = 2, so m^m = 4 and the remaining factor must be m^k − k = 503.
- Then 2^k − k = 503 holds at k = 9 (512 − 9 = 503), so k = 9 (D).
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