The unit's digit in $13^{61}$ is

When dealing with unit's digits in large powers, only the last digit matters.

Calculate the first few powers of 13 and observe their unit's digits:

$13^1 = 13$ → Unit's digit = 3

$13^2 = 169$ → Unit's digit = 9

$13^3 = 2197$ → Unit's digit = 7

$13^4 = 28561$ → Unit's digit = 1

$13^5 = 371293$ → Unit's digit = 3

The unit's digits follow the pattern: 3, 9, 7, 1, 3, 9, 7, 1, ...

The pattern repeats every 4 powers.

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Since the pattern repeats every 4 powers, divide 61 by 4:

$61 \div 4 = 15$ remainder $1$

$61 = 4 \times 15 + 1$

The remainder is 1, which means $13^{61}$ has the same unit's digit as $13^1$.

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From the pattern:

• Power 1 → Unit's digit = 3
• Power 2 → Unit's digit = 9
• Power 3 → Unit's digit = 7
• Power 4 → Unit's digit = 1

Since the remainder is 1, the unit's digit corresponds to position 1 in the pattern.

Therefore, the unit's digit in $13^{61}$ is 3.