3 ELEMENTARYPROBLEMS 5.4Use 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 o416, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, 96.[Hint: Because x² = (50 + x)² (mod 100) and x = (50 – x)² (mod 100), it sufficesexamine the final digits of x² for the 26 values x =Factor the number 211 – 1 by Fermat's factorization method.4. In 1647, Mersenne noted that when a number can be written as a sum of two relativeprime squares in two distinct ways, it is composite and can be factored as follows= a²-0, 1, 2, ..., 25.]-t+ z = -9 + zv = u(ac+ bd)(ac – bd)%D-Use this result to factor the numbers493 = 182 + 132 = 22² + 3²and38025 = 1682 + 992 = 156² + 11725.)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).]6. Factor 13561 with the help of the congruences233² = 32 . 5 (mod 13561)12812 = 2* . 5 (mod 13561)and7. (a) Factor the number 4537 by searching for x such thatx² - k. 4537is the product of small prime powers.(b) Use the procedure indicated in part (a) to factor 14429.[Hint: 1202 – 14429 = -29 and 30032 – 625 · 14429 = –116.18. Use Kraitchik's method to factor the number 20437.

Question

Accepted Answer

Fermat’s Factorization method is based on the representation of an odd integer as the difference of…

the 20 1, 2, 3 Factor the number 211 – 1 by Fermat's factorization method. -