Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
**Problem Statement:** Find the last digit of the decimal (base 10) expansion of \(3^{2823}\). **Explanation:** We need to determine the last digit of the number when expressed in base 10. This is equivalent to finding \(3^{2823} \mod 10\). For problems like these, we can observe the pattern in the last digits of successive powers of 3: - \(3^1 = 3\) (last digit is 3) - \(3^2 = 9\) (last digit is 9) - \(3^3 = 27\) (last digit is 7) - \(3^4 = 81\) (last digit is 1) - \(3^5 = 243\) (last digit is 3) Notice that the last digits repeat every 4 terms: 3, 9, 7, 1. Since the pattern repeats every 4 numbers, we can find the position in the cycle by calculating \(2823 \mod 4\). \(2823 \div 4 = 705\) remainder \(3\). Thus, \(3^{2823}\) corresponds to the third position in the cycle, which has a last digit of 7. Therefore, the last digit of \(3^{2823}\) is 7.
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Transcribed Image Text:**Problem Statement:** Find the last digit of the decimal (base 10) expansion of \(3^{2823}\). **Explanation:** We need to determine the last digit of the number when expressed in base 10. This is equivalent to finding \(3^{2823} \mod 10\). For problems like these, we can observe the pattern in the last digits of successive powers of 3: - \(3^1 = 3\) (last digit is 3) - \(3^2 = 9\) (last digit is 9) - \(3^3 = 27\) (last digit is 7) - \(3^4 = 81\) (last digit is 1) - \(3^5 = 243\) (last digit is 3) Notice that the last digits repeat every 4 terms: 3, 9, 7, 1. Since the pattern repeats every 4 numbers, we can find the position in the cycle by calculating \(2823 \mod 4\). \(2823 \div 4 = 705\) remainder \(3\). Thus, \(3^{2823}\) corresponds to the third position in the cycle, which has a last digit of 7. Therefore, the last digit of \(3^{2823}\) is 7.
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