Modular arithmetic

Japanese school year: Math A

What you learn

You will learn how to perform modular arithmetic by grouping integers that share the same remainder when divided by a fixed number. It is useful for finding remainders of large powers and proving periodic properties. A basic understanding of integer division and remainders is recommended beforehand.

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Key points

This defines modular congruence, stating that integers $a$ and $b$ leave the same remainder when divided by $m$ . This is equivalent to their difference $a - b$ being an integer multiple of $m$ .

a \equiv b \pmod{m} \iff a - b = km \quad (k \in \mathbb{Z})

This is a fundamental rule for modular arithmetic. As long as the modulus $m$ is identical, you can add, subtract, and multiply congruences just like regular equations.

a \equiv b, \; c \equiv d \implies a \pm c \equiv b \pm d, \; ac \equiv bd \pmod{m}

Use this property to find the remainder when a large power is divided by $m$ . Raising both sides to the same power $n$ preserves congruence, letting you simplify the base $a$ to a smaller remainder $b$ first.

a \equiv b \implies a^n \equiv b^n \pmod{m}

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Practice 1Coming soon