Find all solutions of the following congruences:
42 x = 0 (mod 72) .
42 x = 5 (mod 72) .
42 x = 18 (mod 72) .
42^{x} = 36 (mod 72) .
27^{x} = 9 (mod 72) .
Let a, b, and m be integers with m >= 2.
Find the general solution of the congruence
Express the general solution of the congruence
in terms of a known particular solution u.
How many primitive elements are there mod 47 ?
Find all integers x for which
Find the period length of the decimal expansion of the fraction {17}/{47}.
Consider the primes p in the range 3 <= p <= 29. Make a table with three columns as follows: (1) the prime p, (2) the smallest positive primitive root c mod p, and (3) the smallest positive integer k for which one has
Find the smallest prime p for which the smallest positive primitive root c is not also primitive mod p^{2}.
Comment. For a given prime p and a given primitive root u mod p there are p congruence classes mod p^{2} that are congruent to u mod p, and, of these, only one is not also primitive mod p^{2}. If one considers the happenstance that c is the one exception among these to be ``random'', then the ``probability'' of this event is {1}/{p}. Nevertheless, the answer is greater than 1000. This exercise will require machine computation with arbitrary precision integer arithmetic.