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Modular arithmetic is about working with remainders. Imagine you have a bunch of candies and you want to share them equally among friends. The candies that remain after sharing are like the remainders.

In math, modular arithmetic is often written as 'mod'. If we have 12 candies and 5 friends, each friend gets 2 candies (because 12 divided by 5 is 2 with a remainder of 2). We say that 12 mod 5 equals 2, or in mathematical notation: \[12 \mod 5 = 2\]

This idea is handy when working with large numbers because it helps simplify calculations. Instead of dealing with huge numbers, we work with their remainders.

Modular arithmetic is used in many areas like cryptography, computer science, and even in solving everyday problems like scheduling.

Prime numbers are numbers greater than 1 that are only divisible by 1 and themselves. This means they can't be divided evenly by any other number.

Examples of prime numbers include 2, 3, 5, 7, 11, and so on.

Prime numbers are fundamental in number theory because they act like building blocks for all other numbers. Any number can be broken down into a product of prime numbers. For instance, 30 can be written as 2 × 3 × 5.

Fermat's Little Theorem deals specifically with prime numbers. It tells us something very special about any integer and prime number relationship: when you raise an integer to a power related to a prime number, the result has a predictable remainder when divided by that prime.

Exponentiation is about multiplying a number by itself a certain number of times. It is written as a^b where 'a' is the base and 'b' is the exponent.

For example, 2^3 means 2 × 2 × 2, which equals 8. When we deal with exponentiation, especially with large exponents, the numbers can get really big.

Fermat's Little Theorem simplifies exponentiation in modular arithmetic. If you have a prime number p, and another number a that isn't divisible by p, then raising a to the power of p-1 and taking the remainder modulo p will always result in 1. Mathematically, this is written as: \[a^{p-1} \equiv 1 \pmod{p}\]

In the given problem, we used this theorem to find that \[7^{12} \mod 13 = 1\]. This is because 12 is 13-1, saving us from computing large powers by hand.