Problem 9 · 2009 Math Kangaroo
Medium
Number Theory
careful-counting
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
Try small a and compare the digit-lengths of a² and a³.
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Hint 2 of 2
Once a is large enough, a³ always pulls ahead in digit count.
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Approach: check small cases until the lengths must differ
- a=1: 1 and 1 (1 and 1 digits) — same.
- a=2: 4 and 8 (1 and 1) — same; a=3: 9 and 27 (1 and 2) — differ.
- a=4: 16 and 64 (2 and 2) — same; from a=5 onward a³ always has more digits than a².
- Only a = 1, 2, 4 work, so there are 3.
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