= 1564 + 1172 Employ the generalized Fermat method to factor each of the following numbers: (a) 2911 [Hint: 1382 = 672 (mod 2911).] (b) 4573 [Hint: 1772 = 922 (mod 4573).] (c) 6923 [Hint: 2082 = 932 (mod 6923).] Factor 13561 with tl
= 1564 + 1172 Employ the generalized Fermat method to factor each of the following numbers: (a) 2911 [Hint: 1382 = 672 (mod 2911).] (b) 4573 [Hint: 1772 = 922 (mod 4573).] (c) 6923 [Hint: 2082 = 932 (mod 6923).] Factor 13561 with tl
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
5
![### Problems 5.4
1. **Use Fermat’s method to factor each of the following numbers:**
- (a) 2279.
- (b) 10541.
- (c) 340663.
*Hint:* The smallest square just exceeding 340663 is 584².
2. **Prove that a perfect square must end in one of the following pairs of digits:** 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, 96.
*Hint:* Because \( x^2 \equiv (50 + x)^2 \pmod{100} \) and \( x^2 \equiv (50 - x)^2 \pmod{100} \), it suffices to examine the final digits of \( x^2 \) for the 26 values \( x = 0, 1, 2, \ldots , 25 \).
3. **Factor the number \( 2^{11} - 1 \) by Fermat’s factorization method.**
4. **In 1647, Mersenne noted that when a number can be written as a sum of two relatively prime squares in two distinct ways, it is composite and can be factored as follows:**
If \( n = a^2 + b^2 = c^2 + d^2 \), then
\[
n = \frac{(ac + bd)(ac - bd)}{(a + d)(a - d)}
\]
*Use this result to factor the numbers:*
- \( 493 = 18^2 + 13^2 = 22^2 + 3^2 \)
- \( 38025 = 168^2 + 99^2 = 156^2 + 117^2 \)
5. **Employ the generalized Fermat method to factor each of the following numbers:**
- (a) 2911. *Hint:* \( 138^2 \equiv 67^2 \pmod{2911} \).
- (b) 4573. *Hint:* \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F454fa27a-03b0-42a5-b364-d7fc310a1518%2Fnn54wz9.jpeg&w=3840&q=75)
Transcribed Image Text:### Problems 5.4
1. **Use Fermat’s method to factor each of the following numbers:**
- (a) 2279.
- (b) 10541.
- (c) 340663.
*Hint:* The smallest square just exceeding 340663 is 584².
2. **Prove that a perfect square must end in one of the following pairs of digits:** 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, 96.
*Hint:* Because \( x^2 \equiv (50 + x)^2 \pmod{100} \) and \( x^2 \equiv (50 - x)^2 \pmod{100} \), it suffices to examine the final digits of \( x^2 \) for the 26 values \( x = 0, 1, 2, \ldots , 25 \).
3. **Factor the number \( 2^{11} - 1 \) by Fermat’s factorization method.**
4. **In 1647, Mersenne noted that when a number can be written as a sum of two relatively prime squares in two distinct ways, it is composite and can be factored as follows:**
If \( n = a^2 + b^2 = c^2 + d^2 \), then
\[
n = \frac{(ac + bd)(ac - bd)}{(a + d)(a - d)}
\]
*Use this result to factor the numbers:*
- \( 493 = 18^2 + 13^2 = 22^2 + 3^2 \)
- \( 38025 = 168^2 + 99^2 = 156^2 + 117^2 \)
5. **Employ the generalized Fermat method to factor each of the following numbers:**
- (a) 2911. *Hint:* \( 138^2 \equiv 67^2 \pmod{2911} \).
- (b) 4573. *Hint:* \(
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