Modulus refers to the remainder when one integer is divided by another. In mathematics, it is often denoted using the symbol % (in programming) or mod. For example, when you divide 17 by 5, the quotient is 3 and the remainder is 2. Therefore:

$17 \mod 5 = 2$

The modulus operation is particularly useful in number theory, modular arithmetic, and various programming scenarios where you want to find out if numbers divide evenly or determine periodicity.

Important Properties of Modulus

1. Non-negativity: For any integer $a$ and a positive integer $b$, $a \mod b$ is always non-negative and less than $b$.
2. Distributive Property: Modulus distributes over addition and subtraction, e.g., $(a + b) \mod c = [(a \mod c) + (b \mod c)] \mod c$.
3. Even and Odd Check: $x \mod 2 = 0$ indicates that $x$ is even, while $x \mod 2 = 1$ indicates that $x$ is odd.
4. Congruence Relation: Two numbers $a$ and $b$ are congruent modulo $n$ if $(a - b)$ is divisible by $n$. This is written as: $a \equiv b \pmod{n}$
5. Applications: Modulus is crucial in cryptography, cyclic structures, programming algorithms (like hash functions), and clock arithmetic (e.g., 24-hour format).

Would you like to see a detailed example involving modulus, or do you have any other questions?

Related Questions:

1. What is the result of $45 \mod 6$?
2. How does modulus help in determining if a number is divisible by another?
3. Can modulus be negative? Explain with an example.
4. What is modular arithmetic and how is it applied in real-life problems?
5. How does the modulus operation differ from regular division?

Tip: The modulus operator is particularly useful for detecting cycles or periodic patterns in sequences, such as in time calculations or repeating events.