provide a handwritten solution for each question

5. Modular Arithmetic. a) Let u. and m. > 1 be natural numbers with a common divisor d > 1. Prove that the congruence equation rut E 1 (mod m) does not have a solution. b} First solve the equation .132 = :1: for integers, then solve 3:2 E .1: (111ml 5) and .132 E :1: (mod 6). c} Assume a prime number p is of the form 112 + 5 for some natural number n. Prove that the last digit of p must be 1 or 9. d} Suppose the rightmost digit of a natural number n is 7. Prove any prime divisor of n must have the rightmost digit equal to 3 or 7. (Hint: use congruency mod 10.)