The unit's digit of $13^{13}$ is:

When finding the unit's digit of any power, only the unit's digit of the base number matters.

Since $13$ has unit's digit $3$, the pattern of unit's digits when multiplying $3$ by itself repeatedly needs to be found.

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The first few powers of $3$ and their unit's digits:

Power	Calculation	Result	Unit's Digit
$3^1$	$3$	$3$	$3$
$3^2$	$3 \times 3$	$9$	$9$
$3^3$	$9 \times 3$	$27$	$7$
$3^4$	$27 \times 3$	$81$	$1$
$3^5$	$81 \times 3$	$243$	$3$

The pattern repeats: $3 \to 9 \to 7 \to 1 \to 3 \to 9 \to 7 \to 1...$

The cycle length is $4$.

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To find which position in the cycle, divide the power by $4$:

$13 \div 4 = 3$ remainder $1$

The remainder determines the position:

• Remainder $1$ → 1st position in cycle → unit's digit is $3$
• Remainder $2$ → 2nd position in cycle → unit's digit is $9$
• Remainder $3$ → 3rd position in cycle → unit's digit is $7$
• Remainder $0$ → 4th position in cycle → unit's digit is $1$

Since remainder $= 1$, the position is 1st in the cycle.

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The unit's digit of $13^{13}$ is $3$.