Fermat's little theorem
Fermat's little theorem states that if P is
a prime number, then for any integer n ≥1.
Theorem:
(FLT) : Let P is a prime number then,
nP ≡ n(mod p)
Fermat's little theorem
Corollary :
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n,p) = 1
H.C.F
Fermat's little theorem
Example:
(FLT) : Let P is a prime number then,
nP ≡ n(mod p)
n = 5, p = 3
53 ≡ 5(mod 3)
125 ≡ 5(mod 3)
125-5 is divisible by 3
3/125-5
Corollary :
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n,p) = 1
53-1 ≡ 5(mod 3)
52
≡ 5(mod 3)
25 ≡ 5(mod 3)
Fermat's little theorem
Example:
(FLT) : Let P is a prime number then,
nP ≡ n(mod p)
n = 7, p = 2
72 ≡ 7(mod 2)
49 ≡ 7(mod 2)
49-7 is divisible by 2
2/49-7
Corollary :
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n,p) = 1
72-1 ≡ 7(mod 2)
71 ≡ 7(mod 2)
7
≡ 7(mod 2)
Fermat's little theorem
Activity:
1.n = 9, p = 3
2.n = 15, p = 3
3.n = 20, p = 2
Fermat's little theorem
Activity 1:
(FLT) : Let P is a prime number then,
nP ≡ n(mod p)
n = 9, p = 3
93 ≡ 9(mod 3)
729 ≡ 9(mod 3)
729-9 is divisible by 3
3/729-9
Corollary :
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n,p) = 1
93-1 ≡ 1(mod 3)
92 ≡ 1(mod 3)
81 ≡ 1(mod 3)
Fermat's little theorem
Activity 2:
(FLT) : Let P is a prime number then,
nP ≡ n(mod p)
n = 15, p = 3
153
≡ 15(mod 3)
3375 ≡ 15(mod 3)
3375-15 is divisible by 3
3/3375-15
Corollary :
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n,p) = 1
153-1 ≡ 15(mod 3)
152 ≡ 15(mod 3)
225 ≡ 15(mod 3)
Fermat's little theorem
Activity 3:
(FLT) : Let P is a prime number then,
nP ≡ n(mod p)
n = 20, p = 2
202 ≡ 20(mod 2)
400 ≡ 20(mod 2)
400-20 is divisible by 2
2/400-20
Corollary :
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n,p) = 1
202-1 ≡ 20(mod 2)
201 ≡ 20(mod 2)
20
≡ 20(mod 2)
Fermat's little theorem