mod 1537). 2. Frove the following statements: a) For any integer a, the units digit of a? is 0, 1, 4, 5, 6, or 9. (b) Any one of the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 can occur as the units digit of a. c) For any integer a, the units digit of a“ is 0, 1, 5, or 6. (d) The units digit of a triangular number is 0, 1, 3, 5, 6, or 8. decefoanso 099
mod 1537). 2. Frove the following statements: a) For any integer a, the units digit of a? is 0, 1, 4, 5, 6, or 9. (b) Any one of the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 can occur as the units digit of a. c) For any integer a, the units digit of a“ is 0, 1, 5, or 6. (d) The units digit of a triangular number is 0, 1, 3, 5, 6, or 8. decefoanso 099
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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#2 part a and B
![# Educational Content: Modular Arithmetic and Number Theory
### Explanation of Text:
In the provided text, we delve into modular arithmetic, focusing on error detection via check digits and related problems.
The passage discusses how a modular approach (in this case, modulo 10) helps identify errors in sequences of numbers, such as ID numbers, by observing changes in check digits. It demonstrates that these methods can reveal errors like mistyped digits but are not fail-safe, particularly with transpositions of digits.
### Detailed Problem Breakdown:
**PROBLEMS 4.3**
1. **Binary Exponentiation:**
- Compute \(19^{53} \mod 503\) and \(141^{47} \mod 1537\) using the binary exponentiation algorithm. This algorithm efficiently calculates large powers mod a number by utilizing the binary representation of the exponent.
2. **Units Digit Determination:**
- **(a)** For any integer \(a\), identify which digits (0, 1, 4, 5, 6, 9) can be the units digit of \(a^2\).
- **(b)** Determine which digits from 0 to 9 can be the units digit of \(a^3\).
- **(c)** Identify possible units digits (\(0, 1, 5, 6\)) for \(a^4\).
- **(d)** For triangular numbers, discover the units digits \(0, 1, 3, 5, 6, 8\).
3. **Last Digits of Powers:**
- Examine \(9^9\) modulo 10 and observe how these calculations relate to \(9^{99}\) (since \(9^9 \equiv 9^{9+10k} \equiv 89 \mod 100\)).
4. **Divisibility Without Division:**
- Determine if the numbers 176521221 and 149235678 are divisible by 9 or 11 through modular rules, without direct division.
5. **Generalization of a Theorem:**
- For integer \(N\) in base \(b\), using the expansion:
\[
N = a_m b^m + \cdots + a_2 b^2 + a_1 b + a_0
\]
Prove \(b - 1\) divides](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F805ed04c-7627-41e9-b62e-6c0b6aef7157%2Fnqd6xk.jpeg&w=3840&q=75)
Transcribed Image Text:# Educational Content: Modular Arithmetic and Number Theory
### Explanation of Text:
In the provided text, we delve into modular arithmetic, focusing on error detection via check digits and related problems.
The passage discusses how a modular approach (in this case, modulo 10) helps identify errors in sequences of numbers, such as ID numbers, by observing changes in check digits. It demonstrates that these methods can reveal errors like mistyped digits but are not fail-safe, particularly with transpositions of digits.
### Detailed Problem Breakdown:
**PROBLEMS 4.3**
1. **Binary Exponentiation:**
- Compute \(19^{53} \mod 503\) and \(141^{47} \mod 1537\) using the binary exponentiation algorithm. This algorithm efficiently calculates large powers mod a number by utilizing the binary representation of the exponent.
2. **Units Digit Determination:**
- **(a)** For any integer \(a\), identify which digits (0, 1, 4, 5, 6, 9) can be the units digit of \(a^2\).
- **(b)** Determine which digits from 0 to 9 can be the units digit of \(a^3\).
- **(c)** Identify possible units digits (\(0, 1, 5, 6\)) for \(a^4\).
- **(d)** For triangular numbers, discover the units digits \(0, 1, 3, 5, 6, 8\).
3. **Last Digits of Powers:**
- Examine \(9^9\) modulo 10 and observe how these calculations relate to \(9^{99}\) (since \(9^9 \equiv 9^{9+10k} \equiv 89 \mod 100\)).
4. **Divisibility Without Division:**
- Determine if the numbers 176521221 and 149235678 are divisible by 9 or 11 through modular rules, without direct division.
5. **Generalization of a Theorem:**
- For integer \(N\) in base \(b\), using the expansion:
\[
N = a_m b^m + \cdots + a_2 b^2 + a_1 b + a_0
\]
Prove \(b - 1\) divides
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