34 proof note that ii and iii are special cases of i

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Unformatted text preview: as seen in the previous theorem. Theorem 10. Let n ∈ N and a, b, c, d ∈ Z. (i) if a ≡ b(mod n) and c ≡ d(mod n), then a + c ≡ b + d(mod n) and ac ≡ bd(mod n) (ii) if a ≡ b(mod n), a + c ≡ b + c(mod n) and ac ≡ bc(mod n) (iii) if a ≡ b(mod n), then ak ≡ bk (mod n) for any k ∈ N. 34 Proof. Note that (ii) and (iii) are special cases of (i), so that we will only proof (i). Thus let n ∈ N and a, b, c, d ∈ Z. By a previous theorem we know that both a and b have the same remainder when divided by n, say r, and c and d have the sam...
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