Problem 11 · 2009 Math Kangaroo
Medium
Number Theory
place-value
For how many positive whole numbers a do a2 and a3 have the same number of digits?
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Answer: B — 3
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Hint 1 of 2
Test small values: compare digit counts of a^2 and a^3 for a = 1, 2, 3, ...
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Hint 2 of 2
Once a^3 pulls ahead in digit count it never falls back; just count the early matches.
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Approach: check digit counts for small a
- a=1: 1 and 1 (1 digit each). a=2: 4 and 8 (1 each). a=4: 16 and 64 (2 each) - all match.
- a=3 gives 9 and 27 (1 vs 2 digits) - no match; for a>=5 the cube always has more digits.
- So exactly 3 values work (a = 1, 2, 4).
Mark:
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