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Remainder theorem + Binomial

PostPosted: Fri Jun 25, 2010 3:43 pm
by Avishek
(111^11)/(11^11) will give a remainder of? I know that Binomial can be used but can't figure out how

Re: Remainder theorem + Binomial

PostPosted: Mon Jun 28, 2010 9:37 pm
by Amey
111^11 can be written as (110+1)^11
If you expand the binomial, it is -

110^11 +
(11C1)110^10 +
(11C2)110^9 +
... +
1

Observe that each of the terms - 11C1, 11C2.. 11C10 are divisible by 11, as 11 is a prime number. So, the remainder is 1.