Question

Remainder when 110! Is divided by 107^2 (107 square)?

Answer 1

Hi. There is this Wilson formula that you should look up. I will explain it first with an example or two. Eg.1: 2! mod 3 = 2 Eg.2: 4! mod 5 = 4 Eg.3: 6! mod 7 = 6 Where 'mod' simply means 'remainder', unlike the standard division operator '/' that would give 2!/3=0 or 6!/7=102. If you haven't yet noticed a pattern in the example, it is simply: (n-1)! mod n! = n-1 where n is a prime number. Note that the above can also be written as: (n-1)! mod n! = -1. Unlike positive remainders, a negative remainder is used for mathematical simplicity in solving questions and it means that the actual remainder (in this case) is = n-1. (If it was -2 then it means it is n-2) Coming to our question, 110! mod 107^2 = 107 * 106! * 108 * 109 * 110 mod 107^2 = 107 * ( 106! * 108 * 109 * 110 mod 107) = 107 * (-1 * 1 * 2 * 3) = -642 Therefore the remainder is 107^2 - 642 = 10807. Note that -1, 1, 2, 3 are the remainders of 106!, 108, 109, 110 when divided by 107. Cheers!