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# AHH

+1
536
3
+367

1000!/500!^2

Largest n so that 7^n can divide that?

Apr 6, 2019

#1
+114926
+1

$$7^n\div\frac{1000!}{(500!)^2}=N \qquad N\in Z\\$$

500! = 7*14* ....(71*7) * k                    where k is some integer that is not divisale by 7  (each k is a different interger)

500! = 7(1*2*3*.....*71)k = 7*71! * k

(500!)^2 = 49*(71)!^2 *k

1000! = 7*14* ......  142*7 *k

$$\frac{1000!}{(500!)^2}\\ =\frac{7(142!)}{7*71!*7*71!}k\\ =\frac{(142!)}{71!*7*71!}k\\ =\frac{(7*14*21*....20*7)}{7(7*14*21* .... 10*7)*(7*14*21* .... 10*7)}k\\ =\frac{(7*20!)}{7(7*10!)*(7*10!)}k\\ =\frac{(20!)}{7*7*(10!)*(10!)}k\\ =\frac{k}{(10!)*(10!)}\\ =\frac{k}{49}\\$$

So I think the biggest  n must be less than or equal to 2

I could easily have made stupid mistakes though.

Actually I think my answer could easily be rubbish.

Apr 6, 2019
edited by Melody  Apr 6, 2019
edited by Melody  Apr 6, 2019
edited by Melody  Apr 6, 2019
#3
+367
+1

ITs basically 1000! divide by (500!)^2

How many times can that expression be divide by seven?

#2
+1

Not sure if I understand your question!

1 - if 7^n mod (1000! / 500!^2) =0, then there is no integer solution for n.

2 - if (1000! / 500!^2) mod 7^n =0, then n has the following values:

n = -25, -24, -23.........all the way to n =0.

Apr 6, 2019