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.

Any help would be appreciated. THANKS!

Hint 1 : What is remainder when $11!$ is divided by $7$? Hint 2 : This one is simpler. — cosmo5, Mar 14 '21 at 06:40
@SomeGuy Sir I have mentioned that I do not know how to start. — Hussain-Alqatari, Mar 14 '21 at 06:44
@cosmo5 there is no $11!$ in the problem, it is $(1997!+11)!$ — Some Guy, Mar 14 '21 at 06:44
Hint: The entire expression is ultimately just gonna be (some number larger than 7)!. — Chaotic Good, Mar 14 '21 at 06:48
By the linked dupes, the product of any sequence of $,n\ge 7,$ consecutive integers is divisible by $7$ (in fact here $7$ actually occurs as one of the consecutive integers in the prodcut $n!)\ \ $ — Bill Dubuque, Mar 14 '21 at 07:54

score 2 · Answer 1 · answered Mar 14 '21 at 06:47

2

Hint :

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

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

answered Mar 14 '21 at 06:47

cosmo5

10,629

Please strive not to add more dupe answers to dupes of FAQs. – Bill Dubuque Mar 14 '21 at 07:56

Some Guy · Accepted Answer · 2021-03-14T06:58:52.577

2

$((1997!+11)!+29)!$ 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.

edited Mar 14 '21 at 06:58

answered Mar 14 '21 at 06:50

Some Guy

2,687

If suffices to notice that $(1997!+11)!+29 > 7$ – NN2 Mar 14 '21 at 06:55
You mean $7!$? I see what you mean – Some Guy Mar 14 '21 at 06:55
Please strive not to add more dupe answers to dupes of FAQs. – Bill Dubuque Mar 14 '21 at 07:56
@BillDubuque I answered it before it was closed, should I still delete it? – Some Guy Mar 14 '21 at 08:23

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

2 Answers2