HMMT 十一月 2019 · THM 赛 · 第 4 题

HMMT November 2019 — THM Round — Problem 4

To celebrate 2019, Faraz gets four sandwiches shaped in the digits 2, 0, 1, and 9 at lunch. However, the four digits get reordered (but not flipped or rotated) on his plate and he notices that they form a 4-digit multiple of 7. What is the greatest possible number that could have been formed?

解析

To celebrate 2019, Faraz gets four sandwiches shaped in the digits 2, 0, 1, and 9 at lunch. However, the four digits get reordered (but not flipped or rotated) on his plate and he notices that they form a 4-digit multiple of 7. What is the greatest possible number that could have been formed? Proposed by: Milan Haiman Answer: 1092 Note that 2 and 9 are equivalent mod 7. So we will replace the 9 with a 2 for now. Since 7 is a divisor of 21, a four digit multiple of 7 consisting of 2, 0, 1, and 2 cannot have a 2 followed by a 1 (otherwise k we could subtract a multiple of 21 to obtain a number of the form 2 · 10 ). Thus our number either starts with a 1 or has a 0 followed by a 1. We can check that 2201 and 2012 are not divisible by 7. Thus our number starts with a 1. Checking 1220, 1202, and 1022 gives that 1022 is the only possibility. So the answer is 1092.