<p>What is the digit in the unit's place of <span>\((5172)^{11327}\)</span>?</p>
Step-by-Step Solution
Key Concept: The unit digits of powers follow a cycle. For base ending in 2, the cycle is 2, 4, 8, 6 with period 4. Use the remainder when exponent is divided by 4.
<p><strong>Solution:</strong></p><p>The last digit of the base is 2. We need to find the remainder when the exponent 11327 is divided by 4 (the period of unit digits of powers of 2).</p><p><span>$11327 = 4 \times 2831 + 3$</span></p><p>So remainder is 3. Looking at the pattern of unit digits of powers of 2:</p><p><span>$2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 6, 2^5 = 2, ...$</span></p><p>When remainder is 3, the unit digit is <strong>8</strong>.</p>
Correct Answer: 8