If S=1!+2!+3!+…+99!, then the units digit in the value of S is:

1968 AMC 12 Problem 21

Attempt Problem 21 of the 1968 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1968 AMC 12 solutions, or check the answer key.

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21.

S=1!+2!+3!+\cdots+99!,

then the units digit in the value of

S

is:

9

8

5

3

0

Every factorial from

5!

onward ends in zero

Only add the units digits of the first four terms

For

k\ge5,

k!

is divisible by

10.

Only the first four terms matter. Their sum is

1+2+6+24=33,

whose units digit is

3.

Therefore, the correct answer is D.