< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown > Question: Let ~ be the equivalence relation on the set ℤ of integers defined by a~b if a-b is divisible by 5. Let k ∈ Em belong to the equivalence class of m, and l ∈ En belong to the equivalence class of n. Prove that the product kl belongs to the equivalence class Emn. What I've tried: Since k ∈ Em, then m - k = 5x, for x ∈ ℤ. Also, since l ∈ En, then n - l = 5y, for y ∈ ℤ. So, I need to show that m*n - k*l = 5z, for z ∈ ℤ. I've tried multiplying (m-k)*(n-l), but to no avail. So I tried one case. Let m=20, k=10, n=30, and l=5. Well, 20-10=5(2), so 10 ∈ E20, and 30-5=5(5), so 5 ∈ E30 Also, 600-50=5(110), so 50 ∈ E600, which for this case means kl ∈ Emn. Dilemma: I'm convinced the product kl belongs to the equivalence class of m*n always, so I'm having trouble actually finding a systematic way of proving it in general, instead of using a specific case.