<p>What is the digit in the unit's place of <span>\((1008)^{786}\)</span>?</p>
Step-by-Step Solution
Key Concept: Unit digits of powers of 8 cycle with period 4. Use the remainder when dividing the exponent by 4 to find the corresponding unit digit.
<p><strong>Step 1:</strong> Identify the last digit of the base: The last digit of 1008 is 8.</p><p><strong>Step 2:</strong> Determine the period of powers of 8: The unit digits of powers of 8 follow the pattern: 8, 4, 2, 6, 8, 4, 2, 6, ... (period = 4).</p><p><strong>Step 3:</strong> Find the remainder when the exponent is divided by the period: $786 \div 4$ gives remainder 2.</p><p><strong>Step 4:</strong> Use the switch number: When remainder is 2, the switch number is 2, which corresponds to the unit digit 4.</p><p>∴ The digit in the unit's place of $(1008)^{786}$ is <strong>4</strong>.</p>
Correct Answer: 4