HMMT 十一月 2019 · 冲刺赛 · 第 4 题

HMMT November 2019 — Guts Round — Problem 4

[6] In 2019, a team, including professor Andrew Sutherland of MIT, found three cubes of integers which sum to 42: 3 3 3 42 = ( − 8053873881207597 ) + (80435758145817515) + (12602123297335631) One of the digits, labeled by an underscore, is missing. What is that digit?

解析

[6] In 2019, a team, including professor Andrew Sutherland of MIT, found three cubes of integers which sum to 42: 3 3 3 42 = ( − 8053873881207597 ) + (80435758145817515) + (12602123297335631) One of the digits, labeled by an underscore, is missing. What is that digit? Proposed by: Andrew Gu Answer: 4 3 3 3 Let the missing digit be x . Then, taking the equation modulo 10, we see that 2 ≡ − x + 5 + 1 . This 3 simplifies to x ≡ 4 (mod 10), which gives a unique solution of x = 4.