[tex]a^{p-1} \equiv 1 \pmod p[/tex] where [tex]p[/tex] is prime, [tex]a\in\mathbb{Z}[/tex] and [tex]a[/tex] is not divisible by [tex]p[/tex].
[tex]7^{13-1}\equiv 1 \pmod {13}\\7^{12}\equiv 1 \pmod {13}\\\\542=45\cdot12+2\\\\7^{45\cdot 12}\equiv 1 \pmod {13}\\7^{45\cdot 12+2}\equiv 7^2 \pmod {13}\\7^{542}\equiv 49 \pmod{13}[/tex]
Answer:
49 mod 13 = 10.
Step-by-step explanation:
Fermat's little theorem states that
x^p = x mod p where p is a prime number.
Note that 542 = 41*13 + 9 so
7^542 = 7^(41*13 + 9) = 7^9 * (7^41))^13
By FLT (7^41)^13 = 7^41 mod 13
So 7^542 = ( 7^9 * 7(41)^13) mod 13
= (7^9 * 7^41) mod 13
= 7^50 mod 13
Now we apply FLT to this:
50 = 3*13 + 11
In a similar method to the above we get
7^50 = (7^11 * (7^3))13) mod 13
= (7^11 * 7^3) mod 13
= (7 * 7^13) mod 13
= ( 7* 7) mod 13
= 49 mod 13
= 10 (answer).
