Problem 15 · 2022 Math Kangaroo
Hard
Number Theory
factorization
What is the largest common divisor of \(2^{2021}+2^{2022}\) and \(3^{2021}+3^{2022}\)?
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Answer: E — 12
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Hint 1 of 2
Factor each sum: pull out the common power of 2, and of 3.
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Hint 2 of 2
2^2021 + 2^2022 = 3·2^2021 and 3^2021 + 3^2022 = 4·3^2021; now take the gcd.
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Approach: factor then gcd
- 2^2021 + 2^2022 = 2^2021(1+2) = 3·2^2021.
- 3^2021 + 3^2022 = 3^2021(1+3) = 4·3^2021 = 2^2·3^2021.
- Common factors: 2^2 from one side and 3 from the other give gcd = 4·3 = 12.
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