Let p ∈ Z be a prime integer. (a) Show that if p > 3, then p is of the form 6k + 1 or 6k −1 for some k ∈ Z. (b) If p > 5, show that dividing p by 10 can only leave remainders of 1, 3, 7, or 9, and find examples of primes with these remainders.

(1) Let p be a prime number. Show that for any integer a the greatest common divisor of p and a is either p (if pla) or is 1 (if p does not divide a).

In this problem, you can assume the following fact about least common multiples (lcm): FACT: If h, ? are integers such that h|t and ?|t, then lcm(h, ?)|tNow prove the following:Suppose p and q are distinct primes such that ordp a = h and ordq a = k Prove that ordpq a= lcm(h, k) (just check the two conditions of order)