1. Equations mod m
Let a and b be constants in ℤm, where m∈ℕ and m≥2, and let x be a variable. Show that the equation ax=b (mod m) has a unique solution in ℤm if and only if gcd(a,m) = 1.
Recall that for m,n∈ℕ, we write m|n if there exist some k∈ℕ such that n=km.
- Let a∈ℕ, a≠0. Show that m|n if and only if am|an.
Show that m|n if and only if m2|n2.
3. Partial orders
Let R be a partial order on B, and let f:A→B be a function. Define the relation Rf on A by the rule (x,y) ∈ Rf if and only if (f(x),f(y)) ∈ R.
Show that Rf is a partial order if and only if f is injective.