"Mod" is short for "modulo" and refers to the remainder when a number is divided by another number. In this case, it is the remainder when a^5 is divided by 15.
Proving congruence between two numbers is important because it allows us to make predictions and solve equations with large numbers more easily. It also has applications in fields such as cryptography and computer science.
To prove this congruence, we need to show that a^5 and a have the same remainder when divided by 15. This can be done by using the properties of modular arithmetic and manipulating the exponents and bases of the numbers.
If a is not a multiple of 15, we can still prove the congruence by using the fact that a^5 and a will have the same remainder when divided by 15. This is because the remainder of a^5 will depend only on the remainder of a when divided by 15.
Yes, this congruence can hold for other numbers besides 15. In general, we can prove a^5 congruent to a (mod n) for any positive integer n. However, the specific steps and methods used may vary depending on the value of n.