Mathematics
Problem of The Day
2023
88

88

Factorial n, written n!, is defined by: n! = 1 × 2 × 3 × · · · × n.

What is the remainder when 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! + 9! + 10! is divided by 5?

Answer

3

SOLUTIOn

For integer values of n! greater than 4, n! is a multiple of 5. So 5! + 6! + 7! + 8! + 9! + 10! is divisible by 5.

Therefore the remainder when 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! + 9! + 10! is divided by 5 is thesame as when 1!+2!+3!+4! is divided by 5. Now 1!+2!+3!+4! = 1+2+6+24 = 33 = 6×5+3.

Therefore the required remainder is 3.

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