/Solutions

1. Equations mod m

Let a and b be constants in ℤ_m, 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 ℤ_m if and only if gcd(a,m) = 1.

2. Divisibility

Recall that for m,n∈ℕ, we write m|n if there exist some k∈ℕ such that n=km.

1. Let a∈ℕ, a≠0. Show that m|n if and only if am|an.

2. 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 R_f on A by the rule (x,y) ∈ R_f if and only if (f(x),f(y)) ∈ R.

Show that R_f is a partial order if and only if f is injective.