The unit's digit of $2^{2025}$ is

The unit's digit of powers of 2 follows a pattern. Looking at the first few powers:

$2^1 = 2$ → unit's digit is $2$

$2^2 = 4$ → unit's digit is $4$

$2^3 = 8$ → unit's digit is $8$

$2^4 = 16$ → unit's digit is $6$

$2^5 = 32$ → unit's digit is $2$

$2^6 = 64$ → unit's digit is $4$

$2^7 = 128$ → unit's digit is $8$

$2^8 = 256$ → unit's digit is $6$

The pattern of unit's digits repeats every 4 powers: $2, 4, 8, 6, 2, 4, 8, 6, ...$

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To find where $2^{2025}$ falls in this cycle, divide the exponent by $4$:

$2025 \div 4 = 506$ remainder $1$

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The remainder determines the unit's digit:

Remainder $1$ → same unit's digit as $2^1$ → $2$

Remainder $2$ → same unit's digit as $2^2$ → $4$

Remainder $3$ → same unit's digit as $2^3$ → $8$

Remainder $0$ → same unit's digit as $2^4$ → $6$

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Since the remainder is $1$, the unit's digit of $2^{2025}$ is $2$.