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2. (4 points) Verify that 223 is a prime number. Show your work and explain your reasoning.

2. (4 points) Verify that 223 is a prime number. Show your work and explain your reasoning.

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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Certainly! Here is a transcription suitable for an educational website: --- **Exercise 2: Prime Number Verification** **Question (4 points):** Verify that 223 is a prime number. Show your work and explain your reasoning. --- To determine if 223 is a prime number, follow these steps: 1. **Definition of a Prime Number:** A prime number has no positive divisors other than 1 and itself. 2. **Square Root Check:** Estimate the square root of 223. Since \(15^2 = 225\) and \(14^2 = 196\), the square root of 223 is between 14 and 15. 3. **Prime Testing:** Check for divisibility with all prime numbers less than or equal to 15 (i.e., 2, 3, 5, 7, 11, 13). - **Divisibility by 2:** 223 is odd, so it is not divisible by 2. - **Divisibility by 3:** Sum of digits (2 + 2 + 3 = 7) is not divisible by 3. - **Divisibility by 5:** The number does not end in 0 or 5. - **Divisibility by 7, 11, 13:** Perform division: - \(223 \div 7 \approx 31.857\) (not an integer) - \(223 \div 11 \approx 20.2727\) (not an integer) - \(223 \div 13 \approx 17.1538\) (not an integer) Since 223 is not divisible by any prime number up to 15, it is confirmed as a prime number. ---
Transcribed Image Text:Certainly! Here is a transcription suitable for an educational website: --- **Exercise 2: Prime Number Verification** **Question (4 points):** Verify that 223 is a prime number. Show your work and explain your reasoning. --- To determine if 223 is a prime number, follow these steps: 1. **Definition of a Prime Number:** A prime number has no positive divisors other than 1 and itself. 2. **Square Root Check:** Estimate the square root of 223. Since \(15^2 = 225\) and \(14^2 = 196\), the square root of 223 is between 14 and 15. 3. **Prime Testing:** Check for divisibility with all prime numbers less than or equal to 15 (i.e., 2, 3, 5, 7, 11, 13). - **Divisibility by 2:** 223 is odd, so it is not divisible by 2. - **Divisibility by 3:** Sum of digits (2 + 2 + 3 = 7) is not divisible by 3. - **Divisibility by 5:** The number does not end in 0 or 5. - **Divisibility by 7, 11, 13:** Perform division: - \(223 \div 7 \approx 31.857\) (not an integer) - \(223 \div 11 \approx 20.2727\) (not an integer) - \(223 \div 13 \approx 17.1538\) (not an integer) Since 223 is not divisible by any prime number up to 15, it is confirmed as a prime number. ---
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