PUMaC 2019 · 团队赛 · 第 9 题
PUMaC 2019 — Team Round — Problem 9
题目详情
- Find the integer 55 + 3183 + 28969 + 85282 . N N
解析
- Find the integer 55 + 3183 + 28969 + 85282 . Proposed by: Jackson Blitz Answer: 85359 Let k be the desired integer. Then we have 5 5 5 5 4 k < 85282 + 30000 < 85282 + 5 · 100 · 85282 5 5 < (85282 + 100) = (85382) 5 so 85282 is less than k < 85382. Taking the original equation mod 3 gives k ≡ 0 mod 3 so 5 k ≡ 0 mod 3. Taking the original equation mod 5 gives k ≡ 4 mod 5 so k ≡ 4 mod 5.. 5 Taking the original equation mod 8 gives k ≡ 7 mod 8 so k ≡ 7 mod 8. These bounds and equivalences imply k = 85359. 3 N N