What is the remainder when dividing $((1997!+11)!+29)!$ by $7$?

Question

This question appeared in one of the national exams in Saudi Arabia which was on Saturday, November $29, 1997$.

If today is Saturday, what day of the week will it be $((1997!+11)!+29)!$ days from now?

Actually I do not know how to start. I saw a similar problem asking $10^{{10}^{10}}$ days from now, it was a simple one.

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Any help would be appreciated. THANKS!

Answer 1

$((1997!+11)!+29)!$ is your expression. We know that $(1997!+11)!+29$ is a very big integer bigger than $7$, thus $((1997!+11)!+29)!$ is the product of all numbers equal to and less than it, which includes $7$, meaning that the number of days is divisible by $7$, thus the only thing that changes is the week, not the day. For example, if you add $7$ days to Sunday, you'd get Sunday. Thus, if we add a number of days divisible by $7$ to Saturday, we end on Saturday. Thus, adding $((1997!+11)!+29)!$ days, which is divisible by $7$ to Saturday will keep it a Saturday.

Answer 2

Hint :

Which day it is the $n$-th day from today, $n$ satisfying the following? $$n \equiv 0 \pmod 7$$

Notice for $n \ge 7$, $n! \equiv 0 \pmod 7$.