<p>What is the digit in the unit's place of <span>\((13057)^{941120579}\)</span>?</p>
Step-by-Step Solution
Key Concept: Unit digits of powers of 7 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 13057 is 7.</p><p><strong>Step 2:</strong> Determine the period of powers of 7: The unit digits of powers of 7 follow the pattern: 7, 9, 3, 1, 7, 9, 3, 1, ... (period = 4).</p><p><strong>Step 3:</strong> Find the remainder when the exponent is divided by the period: $941120579 \div 4$ gives remainder 3.</p><p><strong>Step 4:</strong> Use the switch number: When remainder is 3, the switch number is 3, which corresponds to the unit digit 3.</p><p>∴ The digit in the unit's place of $(13057)^{941120579}$ is <strong>3</strong>.</p>
Correct Answer: 3