Fermat's Little Theorem states that if p is a prime number and a is an integer such that a is not divisible by p, then a^(p-1) ≡ 1 (mod p). As a result, if you multiply a by (p-1) and divide it by p, you get 1 as the remainder.
Fermat's Little Theorem can be verified using a calculator as follows:
Let's say we want to verify that the theorem holds for a = 3 and p = 7. Using the calculator, we can calculate 3^(7-1) % 7:3(7-1) % 7 = 36 % 7 = 729 % 7 = 9 % 7 = 2
Since the remainder is not 1, Fermat's Little Theorem does not hold for these values of p and a.
If we try a different value of a, such as a = 2, we can verify that the theorem does hold:2(7-1) % 7 = 26 % 7 = 64 % 7 = 1
The remainder is 1, as the theorem predicts.
You can check Fermat's Little Theorem for other values of p and a using a similar process. Make sure you use a calculator or computer program that can handle large exponents and remainders.
Fermat's little theorem is a fundamental result in number theory that states that if p is a prime number and a is any integer, then ap ≡ a (mod p). This means that the remainder of the division of a^p by p is always equal to a.
There are a number of applications and implications of Fermat's little theorem. The following are a few examples: