In S_3 find all elements a such that a^2 = p_0 = identity, and all elements b such that b^3 = p_0 .

Give an example of a pair of numbers that can be used to generate modulo residues to represent the numbers in Z21

Find x ∈ Z so that x^3 ≡ 15 (mod 7^4 ). Use the lifting method.

Find all solutions for x, up to congruence.  x ≡ 3^2017 (mod 100)

Let f and g be any permutations. Show that the order of h=fgf^(-1) is the same as the order of g.

Let a be a natural number such that gcd(a, 10) = 1. Use Euler’s theorem to prove that a^40n+1 and a must have the same ones and tens digits in their base ten expansions

Show that the result of the previous problem does not provide a sufficient condition for the existence of a quadratic residue of n; in other words, find relatively prime integers a and n, with a^((phi(n)/2)) ≡ 1 (mod n), for which the congruence x^(2) ≡ a (mod n) is not solvable.

Find the least residue of 13^456 (mod 31)