Check this useful concepts for Remainder Theorem.

FERMAT’s REMAINDER THEOREMLet p be a prime number and a be a number that is co-prime to p. Then, the remainder obtained when ap-1 is divided by p is 1.

For Example:

Find the remainder when 2100 + 2 is divided by 101?

Solution:

101 is a prime number and 2 is co-prime to 101.

2100 = 2101-1 = This when divided by 101 gives remainder 1 according to Fermat’s Theorem.

Also, 2 when divided by 101 give 2 as remainder.

Hence, Final Remainder = 1 + 2 = 3

WILSON’s REMAINDER THEOREMIf p is a prime number, then (p-1)! + 1 is divisible by p.

For Example:

Find the remainder when 16! Is divided by 17?

Solution:

17 is a prime number.

16! = (16! + 1) – 1 = [(17-1!) +1] – 17+16

According to Wilson’s Theorem [(17-1!) +1] is divisible by 17.

Hence, Final remainder = 16

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