The last digit in the decimal representation of $\left(\frac{1}{5}\right)^{2000}$ is

### Concept & Logic Convert the fraction into its standard decimal base. The behavior of powers applied to decimals determines the terminating trailing digit. ### Step-by-Step Solution * Express the fraction as a decimal: $$ \frac{1}{5} = 0.2 $$ * The given term becomes: $$ (0.2)^{2000} = \left(\frac{2}{10}\right)^{2000} = \frac{2^{2000}}{10^{2000}} $$ * Analyze the cyclicity of the last digit of powers of $2$: * $2^1 = 2$ * $2^2 = 4$ * $2^3 = 8$ * $2^4 = 16$ (ends in 6) The cyclicity length is $4$. * Find the remainder of the exponent when divided by $4$: $$ 2000 \div 4 = 500 \quad (\text{Remainder } 0) $$ A remainder of $0$ corresponds to the 4th position in the cycle, meaning $2^{2000}$ ends exactly in the digit $6$. * Dividing an integer ending in $6$ by a power of $10$ shifts the decimal point but leaves the final non-zero terminating rightmost digit as $6$. ### Exam Strategy & Shortcut Recognize that $\left(\frac{1}{5}\right)^n = (0.2)^n$. The last digit matches the unit digit cyclicity pattern of $2^n$. Because $2000$ is perfectly divisible by $4$, the units digit matches $2^4 = 16$, which ends in $6$. ### Common Pitfall Students often confuse "last digit in the decimal representation" with "first decimal place", picking $5$ because they think of the fraction $5$ dynamically, or selecting $2$ without applying the exponent cycle. ### Final Answer **Therefore, the correct answer is 6.**