SOLUTION: What is the remainder when we divide 1! + 2! + 3! + 4! + ... + 95! by 15?

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Question 1184296: What is the remainder when we divide 1! + 2! + 3! + 4! + ... + 95! by 15?
Found 2 solutions by math_helper, ikleyn:
Answer by math_helper(2461) (Show Source):
You can put this solution on YOUR website!

n! mod 15 = 0 for
Therefore, you just need to look at 1!+2!+3!+4! = 1+2+6+24 = 33
33 mod 15 =

Answer by ikleyn(52788) (Show Source):
You can put this solution on YOUR website!
.

Each term k!, where k >= 5, is divisible by 15 without a remainder, since it contains factors 3 and 5. THEREFORE, the sum 1! + 2! + 3! + . . . + 95! gives the same remainder when is divided by 15, as 1! + 2! + 3! + 4!, which is equal to 1 + 2 + 6 + 24 = 33. The remainder of 33 when divided by 15 is 3. THEREFORE, the ANSWER to the problem's question is 3.

Solved.