What is the remainder when $3^{2025}$ is divided by $10$?

Step 1: Finding the remainder when a number is divided by $10$ is equivalent to finding its unit digit. We analyze the powers of $3$ modulo $10$:

• $3^1\equiv 3(\mathrm{mod}10)$

• $3^2\equiv 9(\mathrm{mod}10)$

• $3^3\equiv 7(\mathrm{mod}10)$

• $3^4\equiv 1(\mathrm{mod}10)$

• $3^5\equiv 3(\mathrm{mod}10)$

Step 2: The unit digits follow a repeating cycle of length $4$: $\{3,9,7,1\}$.

Step 3: Determine the remainder when the exponent $2025$ is divided by the cycle length $4$:

Step 4: The remainder of $3^{2025}(\mathrm{mod}10)$ corresponds to the first element in the cycle:

So the remainder is 3.