Why can 10 be written as -2 modulo 12?

Modular arithmetic is a system of arithmetic for integers in which values wrap around after reaching a certain value called the modulus. The video introduces this concept as central to cryptography.

Conditions: Applies to integers.; A modulus value is needed to define the wrap-around behavior.

In this clip, the modulus is the specific value at which the integer system wraps around and restarts. It acts as the boundary for the cyclic arithmetic.

Conditions: Used within modular arithmetic.; The exact formal notation for modulus is not introduced in this clip.

To verify $a \equiv b \pmod{n}$, divide $a$ by the modulus $n$ using long division and check if the remainder is $b$. If the remainder matches $b$, the congruence is valid.

Conditions: The method is demonstrated for positive dividends and positive moduli.; The speaker calls it a colloquial explanation.

Congruence ($\equiv$) indicates that two numbers have the same remainder when divided by a modulus, whereas equality ($=$) means they are the exact same integer. The video uses congruence in cryptography instead of equality.

Conditions: Used in modular arithmetic contexts.; The modulus is explicitly shown in the examples (e.g., mod 12).

It is true because when 15 is divided by 12, the remainder is 3. This matches the definition of congruence modulo 12 used in the video.

Conditions: Use modulus 12.; Interpret the congruence via the remainder of division by 12.