HMMT 十一月 2020 · GEN 赛 · 第 5 题
HMMT November 2020 — GEN Round — Problem 5
题目详情
- Compute the sum of all positive integers a ≤ 26 for which there exist integers b and c such that a + 23 b + 15 c − 2 and 2 a + 5 b + 14 c − 8 are both multiples of 26.
解析
- Compute the sum of all positive integers a ≤ 26 for which there exist integers b and c such that a + 23 b + 15 c − 2 and 2 a + 5 b + 14 c − 8 are both multiples of 26. Proposed by: David Vulakh Answer: 31 Solution: Assume b and c exist. Considering the two values modulo 13, we find { a + 10 b + 2 c ≡ 2 (mod 13) 2 a + 5 b + c ≡ 8 (mod 13) . Subtracting twice the second equation from the first, we get − 3 a ≡ − 14 (mod 13). So, we have a ≡ 9 (mod 13). Therefore we must either have a = 9 or a = 22. Moreover, both a = 9 and a = 22 yield solutions with b = 0 and c = 3 , 16, depending on the value of a . Thus the answer is 9 + 22 = 31.