Problem 29 · 2017 Math Kangaroo
Stretch
Number Theory
caseworkdivisibility
How many different three-digit numbers ABC are there such that \((A + B)^C\) is a three-digit power of two?
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Answer: E — 21
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Hint 1 of 2
The three-digit powers of two are 128, 256, and 512; each must be written as (A+B) raised to the digit C.
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Hint 2 of 2
For each target, list the ways it is a perfect power base^C, then count the (A,B) digit pairs with A from 1-9.
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Approach: casework over the three three-digit powers of two
- Targets: 128 = 2^7; 256 = 2^8 = 4^4 = 16^2; 512 = 2^9 = 8^3.
- Count digit pairs (A,B) with A>=1: 128 -> A+B=2 (2 numbers); 256 -> A+B=2 (2), A+B=4 (4), A+B=16 (3); 512 -> A+B=2 (2), A+B=8 (8).
- Total = 2 + (2+4+3) + (2+8) = 21 numbers.
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