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Problems 4.4 1. Solve the following linear congruence: (e) 34 x 60(mod 98) (f) 140 x 133(mod 301) (Hint: gcd(140,301) = 7 ) 2. Using congruence, solve the Diophantine equations below: (c) 5 x 53 y 17 4. Solve each of the following sets of simultaneous congruence: (c) x 5(mod 6), x 4(mod11), x 3(mod17) . 11. Prove that the congruence: x a(mod n) and x b(mod m) admit a simultaneous solution if and only if gcd(n, m) a b ; If a solution exists, confirm that it is unique modulo lcm(n, m) . Problems 5.2 12 n + 6 +1. 3. From Fermats theorem deduce that, for any integer n 0, 1311 10. Assuming that a and b are integers not divisible by the prime p , establish the following: (a) If a p b p (mod p), then a b(mod p) (b) If a p b p (mod p), then a p b p (mod p 2 ) (Theres a Hint in the textbook following the question.) 11. Employ Fermats theorem to prove that, if p is an odd prime, then (a) 1 p 1 + 2 2 p + ( p 1) p 1 1(mod p ) (b) 1 p + 2 2 + ( p 1) p 0(mod p) (Theres a Hint in the textbook following the question) 15 Establish the statements below: p (a) If the number M p = 2 1 is composite, where p is a prime, then M p is a pseudo prime. n (b) Every composite number Fn = 2 2 + 1 is a pseudo prime. (n = 0,1,2 )n +1 2 2 (Hint: By Question 21 of Section 2.3, 2 2 implies that 2n n +1

1 2 Fn 1 1; but

Fn 2 2

n +1

1;

Problems 5.3 5. (a) Prove that an integer n > 1 is prime if and only if (n 2)! 1(mod n) . (b) If n is a composite integer, show that (n 1)! 0(mod n), except when n = 4 .