PUMaC 2019 · 个人决赛（B 组） · 第 1 题
PUMaC 2019 — Individual Finals (Division B) — Problem 1
题目详情
- Find all pairs of nonnegative integers ( n, m ) such that 2 = 7 + 9.
解析
- Find all nonnegative integers n and m such that 2 = 7 + 9. Answer : n = 4 , m = 1 is the only solution. Solution : When we look at the equation (mod 3) we get that n is even, since 7 ≡ 1 (mod 3). k k m k Then n = 2 k , where k is a nonnegative integer. Then (2 − 3)(2 + 3) = 7 , so both 2 − 3 k k and 2 + 3 are powers of 7. If k ≥ 3, this is impossible by looking at (mod 8) for 2 − 3. Then we check k = 0 , 1 , 2 to get that k = 2 is a solution. Then n = 4 , m = 1 is the only solution Proposed by Igor Medvedev and Aleksa Milojevi´ c.