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

### Concept & Logic To find the last non-zero digit in a decimal representation of a fraction, it helps to convert the denominator to a power of $10$. This transforms the problem into a simple unit digit cyclicity question. ### Step-by-Step Solution * **Given:** $\left(\frac{1}{5}\right)^{2000}$ * Convert the fraction to a decimal-friendly format by multiplying the numerator and denominator by $2$: $$\left(\frac{1 \times 2}{5 \times 2}\right)^{2000} = \left(\frac{2}{10}\right)^{2000}$$ * This can be written as: $$\frac{2^{2000}}{10^{2000}}$$ * The denominator $10^{2000}$ simply defines the decimal point position (creating $2000$ decimal places). * The last digit of the decimal expansion will therefore be exactly the unit digit of the numerator, $2^{2000}$. * Find the unit digit of $2^{2000}$ using cyclicity. The powers of $2$ repeat their unit digits in a cycle of $4$: $(2, 4, 8, 6)$. * Divide the power $2000$ by the cycle length $4$. The remainder is $0$, meaning it lands on the 4th value in the cycle. * The 4th value in the cyclicity of $2$ is $6$. ### Exam Strategy & Shortcut Recognize immediately that $1/5 = 0.2$. The problem is asking for the last digit of $(0.2)^{2000}$, which is governed purely by the unit digit of $2^{2000}$. Since $2000$ is a multiple of $4$, the unit digit is $2^4 \rightarrow 16 \rightarrow 6$. Solved in 10 seconds. ### Common Pitfall A common mistake is assuming the last digit of a decimal expansion involving fractions like $1/5$ always ends in $5$, confusing division rules with exponent rules. ### Final Answer Therefore, the correct answer is **6**.