How to find this expression $1000! \mod 3^{300}$

Question

How to find this expression $(1000!\mod 3^{300})$?

score 2 · Answer 1 · edited Apr 13 '17 at 12:20

2

Using this, the highest power of $3$ in $1000!>300$

So, the remainder $=0$

edited Apr 13 '17 at 12:20

Community

1

answered Apr 06 '13 at 14:51

lab bhattacharjee

274,582

4

It's even easier than that: Of the factors in $1000!$, there are 333 multiples of 3. And since $333>300$, … – Harald Hanche-Olsen Apr 06 '13 at 14:53

score 2 · Accepted Answer · answered Apr 06 '13 at 14:54

2

$3$ goes into $1000!$ at least $300$ times, since it divides $3, 6, 9, \dots, 900$, and hence $3^{300} \mid 1000!$.

answered Apr 06 '13 at 14:54

Clive Newstead

64,925

score 0 · Answer 3 · answered Apr 06 '13 at 15:11

0

Hint $\rm\ B^\color{#C00}A \mid (AB)!\, =\, 1\cdot\cdot\: B\,\cdot\cdot\: 2B\,\cdot\cdot\ 3B\,\cdots \color{#C00}AB$
thus $\:3^{300}\mid 900!\mid 1000!$

answered Apr 06 '13 at 15:11

Math Gems

19,574

How to find this expression $1000! \mod 3^{300}$

3 Answers3