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Without performing the divisions, determine whether the integers 176521221 and 149235678 are divisible by 9 or 11. 11

Without performing the divisions, determine whether the integers 176521221 and 149235678 are divisible by 9 or 11. 11

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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### Text Transcription for Educational Content #### Overview of Error Detection in Number Systems The text discusses methods of error detection using modulo operations, particularly focusing on detecting errors due to transposing adjacent digits. Using specific weights, errors become apparent if numbers are entered incorrectly. #### Key Concepts - **Modulo 10 Approach**: This method helps in identifying errors by comparing digits in a number. For instance, if the number 81504216 is entered incorrectly as 81504316, the calculation programmed to identify discrepancies will reveal an unexpected digit, indicating an error. - **Limitation of Modulo 10**: This method does not detect all errors, especially those due to adjacent digit transpositions, like in identifiers 81504216 and 81504261 which yield the same check digit. More sophisticated methods using different moduli and weights can prevent this type of error. #### Exercises 1. **Binary Exponentiation Algorithm**: Compute both \(19^{53} \mod 503\) and \(141^{47} \mod 1537\). 2. **Units Digits Proof**: - Show for any integer \(a\), the units digit of \(a^2\) is 0, 1, 4, 5, 6, or 9. - Demonstrate any integer from 0 to 9 can appear as the units digit of \(a^3\). - Prove that for any integer \(a\), the units digit of \(a^4\) is 0, 1, 5, or 6. - Show that the units digit of a triangular number is 0, 1, 3, 5, 6, or 8. 3. **Last Two Digits of \(9^n\)**: - Calculate the last two digits of \(9^{99}\). (Hint: Use \(9^9 \equiv 9 \mod 10\); therefore, \(9^{99} = 9^{9+10k}\), noting that \(9^9 \equiv 89 \mod 100\)). 4. **Divisibility Check**: - Determine whether integers 176521221 and 149235678 are divisible by 9 or 11 without dividing. 5. **Theorem Generalization**: - If an integer \(N\) is represented in base \(b\)
Transcribed Image Text:### Text Transcription for Educational Content #### Overview of Error Detection in Number Systems The text discusses methods of error detection using modulo operations, particularly focusing on detecting errors due to transposing adjacent digits. Using specific weights, errors become apparent if numbers are entered incorrectly. #### Key Concepts - **Modulo 10 Approach**: This method helps in identifying errors by comparing digits in a number. For instance, if the number 81504216 is entered incorrectly as 81504316, the calculation programmed to identify discrepancies will reveal an unexpected digit, indicating an error. - **Limitation of Modulo 10**: This method does not detect all errors, especially those due to adjacent digit transpositions, like in identifiers 81504216 and 81504261 which yield the same check digit. More sophisticated methods using different moduli and weights can prevent this type of error. #### Exercises 1. **Binary Exponentiation Algorithm**: Compute both \(19^{53} \mod 503\) and \(141^{47} \mod 1537\). 2. **Units Digits Proof**: - Show for any integer \(a\), the units digit of \(a^2\) is 0, 1, 4, 5, 6, or 9. - Demonstrate any integer from 0 to 9 can appear as the units digit of \(a^3\). - Prove that for any integer \(a\), the units digit of \(a^4\) is 0, 1, 5, or 6. - Show that the units digit of a triangular number is 0, 1, 3, 5, 6, or 8. 3. **Last Two Digits of \(9^n\)**: - Calculate the last two digits of \(9^{99}\). (Hint: Use \(9^9 \equiv 9 \mod 10\); therefore, \(9^{99} = 9^{9+10k}\), noting that \(9^9 \equiv 89 \mod 100\)). 4. **Divisibility Check**: - Determine whether integers 176521221 and 149235678 are divisible by 9 or 11 without dividing. 5. **Theorem Generalization**: - If an integer \(N\) is represented in base \(b\)
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