"What is the last non-zero digit of((⋯(((1!)!+2!)!+3!)!+⋯)!+1992!)!?" solved and explained

Jaylan Chung

Answered question

2022-03-17

What is the last non-zero digit of

((\dots (((1!)! + 2!)! + 3!)! + \dots)! + 1992!)!

sexec5m

Beginner2022-03-18Added 4 answers

Step 1
Let

a_{n}

denote the last nonzero digit of n!; we need only consider

n \geq 5

Prove

a_{n} \in S = {2,, 4,, 6,, 8}

Prove

a_{10 k + 5} = a_{10 k + 14}

Determine the order of the permutation

σ

of S for which

a_{10 k + 15} = σ (a_{10 k + 14})

Find a fairly small upper bound p on the period of

a_{n}

Convince yourself

a_{p},, \dots,, a_{2 p - 1}

is a period.
Explain why the original problem's answer is

a_{p}

then compute it.

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