Using Fermat’s theorem, find 4^225 mod 13.
Fermat's Little Theorem states that if "p" is a prime number and "a" is any positive integer not divisible by "p," then:
a(p-1) ≡ 1 (mod p)
In this case, we have "p" as 13 and "a" as 4. Therefore, according to Fermat's theorem:
4(13-1) ≡ 1 (mod 13)
412 ≡ 1 (mod 13)
Now, we can use this result to find 4225 mod 13 by simplifying the exponent:
4225 = 4(12 * 18 + 9)
Since 412 ≡ 1 (mod 13), we can substitute this into the equation:
4225 ≡ (412)18 * 49 ≡ 118 * 49 ≡ 49 (mod 13)
Now, need to calculate 49 mod 13:
49 = (43)3 = 643 ≡ 13 ≡ 1 (mod 13)
Therefore, 4225 mod 13 ≡ 1.
In other words, 4 raised to the power of 225, when taken modulo 13, leaves a remainder of 1.