PUMaC 2013 · 团队赛 · 第 4 题

PUMaC 2013 — Team Round — Problem 4

Find the sum of all positive integers m such that 2 can be expressed as a sum of four factorials (of positive integers). 4 Note: The factorials do not have to be distinct. For example, 2 = 16 counts, because it equals 3! + 3! + 2! + 2!.

解析

Find the sum of all positive integers m such that 2 can be expressed as a sum of four factorials (of positive integers). 4 Note: The factorials do not have to be distinct. For example, 2 = 16 counts, because it equals 3! + 3! + 2! + 2!. 1 SOLUTION: Clearly, m ≥ 2. WLOG assume a ≤ b ≤ c ≤ d . m If a = 1, then 4 | 2 suggests b = 1 and c ≤ 3. If ( a, b, c ) = (1 , 1 , 1), we must have d = 1, corresponding to m = 2. If ( a, b, c ) = (1 , 1 , 2), taking modulo 8 yields d ≤ 3, which has no solution. If ( a, b, c ) = (1 , 1 , 6), taking modulo 16 yields d ≤ 5. There are two solutions ( a, b, c, d ) = (1 , 1 , 6 , 24) or (1 , 1 , 6 , 120), corresponding to m = 5 or m = 7. m Similarly, if a = 2, then 4 | 2 suggests b = 2 , 3. If ( a, b ) = (2 , 2), we have the solutions ( a, b, c, d ) = (2 , 2 , 2 , 2) or (2 , 2 , 6 , 6), corresponding to m = 3 or m = 4. If ( a, b ) = (2 , 3), then c = 3 , 4 , 5, each of which yields no solution. m If a ≥ 3, then 3 | a ! + b ! + c ! + d ! = 2 , which has no solution. ∑ So the answer is m = 2 + 3 + 4 + 5 + 7 = 21. ANSWER: 21