The unit's digit of $13^{2003}$ is

### Concept & Logic The unit digit of a number raised to a large power follows a predictable repeating pattern called cyclicity. The unit digit of $x^n$ only depends on the unit digit of $x$. ### Step-by-Step Solution **Given:** Find the unit digit of $13^{2003}$. **Calculation:** * The unit digit of $13^{2003}$ is strictly determined by the unit digit of the base, which is $3$. We need the unit digit of $3^{2003}$. * The powers of $3$ follow a cyclicity of $4$: * $3^1 = 3$ * $3^2 = 9$ * $3^3 = 27 \rightarrow 7$ * $3^4 = 81 \rightarrow 1$ * Divide the exponent $2003$ by the cyclicity $4$ to find the remainder: $$2003 \pmod 4 = 3$$ * A remainder of $3$ means the cycle lands on the $3$rd position in the sequence, which corresponds to $3^3$. * The unit digit of $3^3$ is $7$. ### Exam Strategy & Shortcut To quickly find $2003 \pmod 4$, just look at the last two digits of the exponent. $03 / 4$ leaves a remainder of $3$. So, $3^3$ gives a unit digit of $7$. This takes less than $10$ seconds. ### Common Pitfall A common mistake is using the entire base $13$ for multiplication cycles instead of stripping it down to its unit digit $3$, wasting time on unnecessary mental arithmetic. ### Final Answer **Therefore, the correct answer is 7.**