Fermat's little Theorem
If p is a prime then for any integer a we have
ap = a modulo p.
i.e. If p is a prime and n is an integer then np–n is divisible by p.
Example : 7 is a prime so n 7 – n is divisible by 7 .
For n = 2 : 2 7 – 2 = 128 – 2 = 126 is divisible by 7
Questions
Q1 : what is the reminder when 1139 is divided by 19
Q2. Find the reminder when 591 is divided by 91
Q3. Find the following reminders
a. 757575 is divided by 37
b. 2100 is divided by 101
c. 20 51 97 is divided by 17
Corollary :
nq – n is divisible by q where q is a prime number or product of two prime numbers.
Important Points:
1. If P be a prime number such that ap – bp is divisible by p, then it is also divisible by p2
2. If an integer n is greater than 2, then the equation an + bn = cn has no solutions in non-zero integers a, b, and c. (Fermat’s Last Theorem)
3. If p be prime and a is prime to p, then a(p-1) – 1 is multiple of p.
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sir,
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himanshu
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As per fermat's little theorem (1119 - 11)/19 = k
=> 1119 - 11 = 19k
=> 1119 = 19k + 11 (squaring it )
=> 1138 = some multiple of 19 + 121
So reminder of 1138 divided by 19 is 7
=> reminder of 1139 divided by 19 is 77
= reminder of 1139 divided by 19 is 1
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