# 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.

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

^{2}|n^{2}.

# 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.