PUMaC 2009 · 数论（B 组） · 第 3 题

PUMaC 2009 — Number Theory (Division B) — Problem 3

You are given that 17! = 355687 ab 8096000 for some digits a and b . Find the two-digit number ab that is missing above.

解析

You are given that 17! = 355687 ab 8096000 for some digits a and b . Find the two-digit number ab that is missing above. Solution. 42. First note that the number is divisible by 11 as well as 9. We simply apply the divisibility criteria for these two numbers, and immediately obtain two simultaneous linear equations: 9 | 34 + a + b + 23 and 11 | (16 + a + 17) − (18 + b + 6) which give the following possibilities: ( a + b ) ∈ { 6 , 15 } and ( a − b ) ∈ {− 9 , 2 , 13 } , where a and b are digits. Now, a − b = − 9 iff b = 9 , a = 0, which does not satisfy any of the relations on a + b . So, that possibility is eliminated. Furthermore, note that a + b and a − b added together gives an even number 2 a , so the parities of a + b and a − b must be equal. It follows that either we have a + b = 6 , a − b = 2, or a + b = 15 , a − b = 13. Solving the first equation gives 1 ( a, b ) = (4 , 2) and the second gives ( a, b ) = (14 , 1), but since a and b are digits, it follows that our required solution is 42.