A modular congruence is a mathematical relationship that expresses the idea that two integers leave the same remainder when divided by a modulus.

The general modular congruence formula is:

$a \equiv b \pmod{m}$

This means that a and b have the same remainder when divided by m, or equivalently, their difference is divisible by m. In other words:

$m \, | \, (a - b)$

Where $m$ is the modulus and $|$ denotes divisibility.

Examples:

• $17 \equiv 5 \pmod{12}$ means that when 17 and 5 are divided by 12, they leave the same remainder, i.e., 17 and 5 differ by 12, which is divisible by 12.
• $23 \equiv 5 \pmod{18}$ means that 23 and 5 leave the same remainder when divided by 18.

Properties of Modular Congruences:

1. Reflexivity: $a \equiv a \pmod{m}$ (Any number is congruent to itself modulo any modulus).
2. Symmetry: If $a \equiv b \pmod{m}$, then $b \equiv a \pmod{m}$.
3. Transitivity: If $a \equiv b \pmod{m}$ and $b \equiv c \pmod{m}$, then $a \equiv c \pmod{m}$.
4. Addition: If $a \equiv b \pmod{m}$ and $c \equiv d \pmod{m}$, then $(a + c) \equiv (b + d) \pmod{m}$.
5. Multiplication: If $a \equiv b \pmod{m}$ and $c \equiv d \pmod{m}$, then $(a \cdot c) \equiv (b \cdot d) \pmod{m}$.

Use in Solving Problems:

Modular arithmetic is crucial for solving problems in number theory, cryptography, and computer science (e.g., public key cryptography, hashing, etc.).

Would you like to explore a specific example or a related concept?

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Related questions:

1. How do you solve modular congruences in equations?
2. What are the properties of modular inverses in modular arithmetic?
3. How does the Chinese Remainder Theorem work with modular congruences?
4. What is the use of modular arithmetic in cryptography?
5. How does Fermat's Little Theorem relate to modular arithmetic?

Tip: Always simplify the modular expressions by finding the remainder before applying the formula, especially in larger numbers.