Find the remainder when $1! + 2! + 3! + \dots + 100!$ is divided by 30.
Sum (n!)(n=1 to 100) mod 30 = 33 Remainder
+118723 Find the remainder when $1! + 2! + 3! + \dots + 100!$ is divided by 30.
I am still new to modular arithemtic but this is what I am thinking that this is the same as
the remainder when 1! is divided by 30 + the reminder when 2! is divided by 30 +...... the reminder when 100! is divided by 30.
And then after all these reminders are added together divide them by 30 and find the remainder of the original question.
1!=1 and 1mod30=1
2!=2 and 2mod30=2
3!=6 and 6mod30=6
4!=6*4 and 24mod30=24
5!=24*5=120 120/30= 4 so 5!=0 mod 30
6!=120*6 120*6/30=4*6=24 so 6!=0 mod 30
all the others up to 100! will equal 0 mod 30
so
(1! + 2! + 3! + \dots + 100!) mod30 = (1+2+6+24) mod30 = 33mod30 = 3
