What is the remainder when 1!+2!+3!+4!+5!+.......+50! is divided by 5!?

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What is the remainder when 1!+2!+3!+4!+5!+.......+50! is divided by 5!

My Approach

$1$+$2$+$6$+$24$+$5$!/$5$!+$6 . 5$!/$5$!+$7$ .$6$ . $5$!/$5$!....so on

$33$+$1$+$6$+$42$+......

I am not getting the correct answer as the solution is getting complex.

Can anyone guide me how to approach the problem?

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2 Answers

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Hint: Terms of $5!$ onwards are divisble by $5!$, so you only need the remainer of $1! +2!+3!+4!$.

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As Lee said $5!$ onwards all are divisible by $5!$ so we need remainder of($1!+2!+3!+4!$) 33 so remainder is 33.

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