MODULAR ARITHMETIC

The idea

We write $a \equiv b \pmod{n}$ to mean "$a$ and $b$ have the same remainder when divided by $n$." Equivalently, $n$ divides $a - b$.

For example, $17 \equiv 2 \pmod{5}$ because $17 = 3\cdot 5 + 2$.

Why it matters on AMC10

Many problems ask for a units digit, a remainder, or a divisibility property. Working mod 10 (units digit) or mod 9 (digit sum) turns ugly arithmetic into small-number arithmetic.

Rules you can use

Modular arithmetic respects addition, subtraction, and multiplication:

$a \equiv b \pmod n \;\text{and}\; c \equiv d \pmod n \implies a + c \equiv b + d \pmod n$ $\text{and}\; a\cdot c \equiv b\cdot d \pmod n.$

Division is trickier and only works if $\gcd(\text{divisor}, n) = 1$.

Worked example

What is the units digit of $7^{2024}$?

Work mod 10. The units digit of $7^k$ cycles: $7, 9, 3, 1, 7, 9, 3, 1, \ldots$, with period 4. Since $2024 = 4 \cdot 506$, we're at the end of a cycle, so $7^{2024} \equiv 1 \pmod{10}$. The units digit is $1$.