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Find the last digit of the decimal (base 10) expansion of 32823

Find the last digit of the decimal (base 10) expansion of 32823

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**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.
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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