What is the tens digit in the sum 7! + 8! + 9! + … + 2006!

You can get the tens digit of any number n by computing the quotient

(n (mod 100)) / 10

and ignoring the remainder.

Taking the given sum (mod 100) gives

7! + 8! + … + 2006! ≡ 7! + 8! + 9! (mod 100)

since the last 1997 terms (i.e. 10! up to 2006!) in the sum are multiples of 100. That is,

• every term beyond 100! is obviously a multiple of 100

• every term beyond 25! contains a factor of both 4 and 25

• every term beyond 10! contains two factors each of both 2 and 5 (i.e. every factorial term contains 4, 5, and 10)

The remaining sum is easy to compute by hand:

7! + 8! + 9! = 7! (1 + 8 + 8 × 9) = 5040 × 81 = 408,240

so the tens digit is 4.

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