(d) An in 8. For any integer a, show that a² – a +7 ends in one of the digits 3, 7, or 9. - mainder when 44444444 is divided by 9
(d) An in 8. For any integer a, show that a² – a +7 ends in one of the digits 3, 7, or 9. - mainder when 44444444 is divided by 9
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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#8
![**Elementary Number Theory**
1. Criteria for Divisibility
- **(b)** Provide criteria for the divisibility of \( N \) by 3 and 8 based on its digits written in base 9.
- **(c)** Is the integer \( (447836)_9 \) divisible by 3 and 8?
2. Problem Solving
- **6.** Use modulo 9 or 11, find the missing digits in the calculations below:
- **(a)** \( 51840 \cdot 273581 = 1418243x040 \)
- **(b)** \( 2x99561 = [3(523 + x)]^2 \)
- **(c)** \( 2784x = x \cdot 5569 \)
- **(d)** \( 512 \cdot 1x53125 = 10000000000 \)
3. Establish Divisibility Criteria
- **7.**
- **(a)** An integer is divisible by 2 if and only if its units digit is 0, 2, 4, 6, or 8.
- **(b)** An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.
- **(c)** An integer is divisible by 4 if and only if the number formed by its tens and units digits is divisible by 4.
- *[Hint: \(10^k \equiv 0 \pmod{4}\) for \( k \geq 2.\)]*
- **(d)** An integer is divisible by 5 if and only if its units digit is 0 or 5.
4. Number Properties and Proofs
- **8.** Show that for any integer \( a \), \( a^2 - a + 7 \) ends in one of the digits 3, 7, or 9.
- **9.** Prove that when \( 4444^{4444} \) is divided by 9, it yields a certain result.
- *[Hint: Observe that \( 2^3 \equiv -1 \pmod{9}.\)]*
- **10.** Prove that no integer whose digits add up to 15 can be a square or a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2Fb478fa28-e1ab-4f76-bf96-cfe8d2f8b146%2Fuxijvzb.jpeg&w=3840&q=75)
Transcribed Image Text:**Elementary Number Theory**
1. Criteria for Divisibility
- **(b)** Provide criteria for the divisibility of \( N \) by 3 and 8 based on its digits written in base 9.
- **(c)** Is the integer \( (447836)_9 \) divisible by 3 and 8?
2. Problem Solving
- **6.** Use modulo 9 or 11, find the missing digits in the calculations below:
- **(a)** \( 51840 \cdot 273581 = 1418243x040 \)
- **(b)** \( 2x99561 = [3(523 + x)]^2 \)
- **(c)** \( 2784x = x \cdot 5569 \)
- **(d)** \( 512 \cdot 1x53125 = 10000000000 \)
3. Establish Divisibility Criteria
- **7.**
- **(a)** An integer is divisible by 2 if and only if its units digit is 0, 2, 4, 6, or 8.
- **(b)** An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.
- **(c)** An integer is divisible by 4 if and only if the number formed by its tens and units digits is divisible by 4.
- *[Hint: \(10^k \equiv 0 \pmod{4}\) for \( k \geq 2.\)]*
- **(d)** An integer is divisible by 5 if and only if its units digit is 0 or 5.
4. Number Properties and Proofs
- **8.** Show that for any integer \( a \), \( a^2 - a + 7 \) ends in one of the digits 3, 7, or 9.
- **9.** Prove that when \( 4444^{4444} \) is divided by 9, it yields a certain result.
- *[Hint: Observe that \( 2^3 \equiv -1 \pmod{9}.\)]*
- **10.** Prove that no integer whose digits add up to 15 can be a square or a
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