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 Question from Kaylyn, a student: determine the tens digit for the expression: 0! + 1! + 2! + 3! + 4! .............+9999! + 10000!

We have two responses for you

Kaylyn,

I'll show you how to determine the units digit of 0! + 1! + 2! + 3! + 4! .............+9999! + 10000! and then you can use a similar argument to determine the tens digit. I start by doing some arithmetic.

0! = 1
1! = 1
2! = 2
3! = 3 × 2! = 6
4! = 4 × 3! = 24
5! = 5 × 4! = 120 = 12 × 10
6! = 6 × 5! = 720 = 72 × 10
7! = something × 10
and so on.

Thus for every integer n larger than 4, n! is a multiple of 10 and hence has a units digit of 0. Thus the units digit of 0! + 1! + 2! + 3! + 4! .............+9999! + 10000! is the units digit of the sum 0! + 1! + 2! + 3! + 4! = 1 + 1 + 2 + 6 + 24 = 34. Hence the units digit of 0! + 1! + 2! + 3! + 4! .............+9999! + 10000! is 4.

I hope this helps,
Penny

Work out (say) the first twelve factorials. You may use a calculator, but should do it by multiplying each factorial by the next natural number: as, 1x2 = 2, 2x3 = 6, 6x4 = 24, 24 x 5 = 120.

You should already see a pattern; if not, compute the sums

0! = 1

1 + 1 = 2
2 + 2 = 4
4 + 6 = 10
10 + 24 = 34

and you will surely see the pattern that will let you answer the question.

Good Hunting!
RD

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