4. In 1647, Mersenne noted that when a number can be written as a sum of two relativ prime squares in two distinct ways, it is composite and can be factored as follows n = a² + b² = c² + d², then (ас + bd)(ac -bd) n = (a+d)(a – d) - Use this result to factor the numbers 493 = 182 + 132 = 22² + 3² and 38025 = 1682 + 992 = 156² + 1172 %3D %3D 5 JEmplou th
4. In 1647, Mersenne noted that when a number can be written as a sum of two relativ prime squares in two distinct ways, it is composite and can be factored as follows n = a² + b² = c² + d², then (ас + bd)(ac -bd) n = (a+d)(a – d) - Use this result to factor the numbers 493 = 182 + 132 = 22² + 3² and 38025 = 1682 + 992 = 156² + 1172 %3D %3D 5 JEmplou th
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
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![**Elementary Number Theory: 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%2F85d7b195-78cd-47b4-8f38-7254d1dd14fd%2Ftcgzqat.jpeg&w=3840&q=75)
Transcribed Image Text:**Elementary Number Theory: 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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