<p>What is the digit in the unit's place of <span>\((2419)^{111213}\)</span>?</p>
Step-by-Step Solution
Key Concept: Unit digits of powers of 9 cycle with period 2. Use the remainder when dividing the exponent by 2 to find the corresponding unit digit.
<p><strong>Step 1:</strong> Identify the last digit of the base: The last digit of 2419 is 9.</p><p><strong>Step 2:</strong> Determine the period of powers of 9: The unit digits of powers of 9 follow the pattern: 9, 1, 9, 1, 9, ... (period = 2).</p><p><strong>Step 3:</strong> Find the remainder when the exponent is divided by the period: $111213 \div 2$ gives remainder 1.</p><p><strong>Step 4:</strong> Use the switch number: When remainder is 1, the switch number is 1, which corresponds to the unit digit 9.</p><p>∴ The digit in the unit's place of $(2419)^{111213}$ is <strong>9</strong>.</p>
Correct Answer: 9