HMMT 十一月 2011 · 冲刺赛 · 第 23 题

HMMT November 2011 — Guts Round — Problem 23

[ 12 ] Let N = 5 AB 37 C 2, where A, B, C are digits between 0 and 9, inclusive, and N is a 7-digit positive integer. If N is divisible by 792, determine all possible ordered triples ( A, B, C ).

解析

[ 12 ] Let N = 5 AB 37 C 2, where A, B, C are digits between 0 and 9, inclusive, and N is a 7-digit positive integer. If N is divisible by 792, determine all possible ordered triples ( A, B, C ). 3 2 Answer: (0 , 5 , 5) , (4 , 5 , 1) , (6 , 4 , 9) First, note that 792 = 2 × 3 × 11 . So we get that 8 | N ⇒ 8 | 7 C 2 ⇒ 8 | 10 C + 6 ⇒ C = 1 , 5 , 9 9 | N ⇒ 9 | 5 + A + B + 3 + 7 + C + 2 ⇒ A + B + C = 1 , 10 , 19 11 | N ⇒ 11 | 5 − A + B − 3 + 7 − C + 2 ⇒ − A + B − C = − 11 , 0 Adding the last two equations, and noting that they sum to 2 B , which must be even, we get that B = 4 , 5. Checking values of C we get possible triplets of (0 , 5 , 5), (4 , 5 , 1), and (6 , 4 , 9).