MS 5.4 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.]

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
Question

1

**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 \(x^2 \equiv (50 + x)^2 \mod 100\) and \(x^2 \equiv (50 - x)^2 \mod 100\), 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: \(138^2 \equiv 67^2 \mod 2911\).]
   - (b) 4573 [*Hint: \(177^2 \equiv 92^
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 \(x^2 \equiv (50 + x)^2 \mod 100\) and \(x^2 \equiv (50 - x)^2 \mod 100\), 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: \(138^2 \equiv 67^2 \mod 2911\).] - (b) 4573 [*Hint: \(177^2 \equiv 92^
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