3rd Centromerican 2001 Problem B1 Find the smallest n such that the sequence of positive integers a 1 , a 2 , ... , a n has each term ≤ 15 and a 1 ! + a 2 ! + ... + a n ! has last four digits 2001.
We find that the last 4 digits are as follows: 1! 1, 2! 2, 3! 6, 4! 24, 5! 120, 6! 720, 7! 5040, 8! 320, 9! 2880, 10! 8800, 11! 6800, 12! 1600, 13! 800, 14! 1200, 15! 8000. Only 1! is odd, so we must include it. None of the others has last 4 digits 2000, so we need at least three factorials. But 13! + 14! + 1! works. 3rd OMCC 2001 © John Scholes jscholes@kalva.demon.co.uk 4 December 2003 Last corrected/updated 4 Dec 03