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Find the last digit in the decimal expansion of 719064

Find the last digit in the decimal expansion of 719064

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 in the decimal expansion of \(7^{19064}\). **Solution Explanation:** To find the last digit of a number, we are essentially finding the remainder when the number is divided by 10. Thus, this problem can be solved by determining \(7^{19064} \mod 10\). **Step-by-step Approach:** 1. **Cycle of Powers Modulo 10:** - We need to observe the pattern of the last digits of consecutive powers of 7. - Calculate: - \(7^1 = 7 \equiv 7 \mod 10\) - \(7^2 = 49 \equiv 9 \mod 10\) - \(7^3 = 343 \equiv 3 \mod 10\) - \(7^4 = 2401 \equiv 1 \mod 10\) - Notice that \(7^4 \equiv 1 \mod 10\). 2. **Identifying the Cycle:** - The powers of 7 modulo 10 repeat every 4 numbers: 7, 9, 3, 1. - This means that every fourth power will end in the same digit 1. 3. **Applying the Cycle to \(7^{19064}\):** - Divide the exponent by the length of the cycle: \(19064 \div 4 = 4766\) remainder \(0\). - A remainder of 0 means \(7^{19064}\) aligns with the last number in the cycle, which is 1. **Conclusion:** Thus, the last digit of \(7^{19064}\) is **1**.
Transcribed Image Text:**Problem Statement:** Find the last digit in the decimal expansion of \(7^{19064}\). **Solution Explanation:** To find the last digit of a number, we are essentially finding the remainder when the number is divided by 10. Thus, this problem can be solved by determining \(7^{19064} \mod 10\). **Step-by-step Approach:** 1. **Cycle of Powers Modulo 10:** - We need to observe the pattern of the last digits of consecutive powers of 7. - Calculate: - \(7^1 = 7 \equiv 7 \mod 10\) - \(7^2 = 49 \equiv 9 \mod 10\) - \(7^3 = 343 \equiv 3 \mod 10\) - \(7^4 = 2401 \equiv 1 \mod 10\) - Notice that \(7^4 \equiv 1 \mod 10\). 2. **Identifying the Cycle:** - The powers of 7 modulo 10 repeat every 4 numbers: 7, 9, 3, 1. - This means that every fourth power will end in the same digit 1. 3. **Applying the Cycle to \(7^{19064}\):** - Divide the exponent by the length of the cycle: \(19064 \div 4 = 4766\) remainder \(0\). - A remainder of 0 means \(7^{19064}\) aligns with the last number in the cycle, which is 1. **Conclusion:** Thus, the last digit of \(7^{19064}\) is **1**.
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