Problems 4.41. Solve the following linear congruence:(e) (f) (Hint: )

2. Using congruence, solve the Diophantine equations below:(c)

4. Solve each of the following sets of simultaneous congruence:(c) .

11. Prove that the congruence: and admit a simultaneous solution if and only if ; If a solution exists, confirm that it is unique modulo .

Problems 5.23. From Fermat’s theorem deduce that, for any integer .

10. Assuming that and are integers not divisible by the prime , establish the following:(a) If then (b) If then (There’s a “Hint” in the textbook following the question.)

11. Employ Fermat’s theorem to prove that, if is an odd prime, then(a) (b) (There’s a “Hint” in the textbook following the question)

15 Establish the statements below:(a) If the number is composite, where is a prime, then is a pseudo prime.(b) Every composite number is a pseudo prime. (Hint: By Question 21 of Section 2.3, implies that but

Problems 5.35. (a) Prove that an integer is prime if and only if . (b) If is a composite integer, show that except when .