Well-definition of addition modulo n

Well-definition of addition modulo n Prove each of the following important statements about addition modulo n, where n is any integer greater than or equal to 2. When applicable, explain how it has changed your life. 1. If a ≡ 1 (mod 3), then a + 1 ≡ 2 (mod 3). 2. If a ≡ 2 (mod 3), then a + 5 ≡ 1 (mod 3). 3. If a ≡ 3 (mod 4), then a + 1 ≡ 0 (mod 4) 4. If 3a ≡ 2 (mod 7) and 2a ≡ 6 (mod 7), then a ≡ 3 (mod 7). 5. If a ≡ b (mod n), then a + 1 ≡ b + 1 (mod n). 6. If a ≡ b (mod n), then for any integer k, a + k ≡ b + k (mod n). ∞. If a1 ≡ b1 (mod n) and a2 ≡ b2 (mod n), then a1 + a2 ≡ b1 + b2 (mod n). The last statement (problem “∞”) is called the well-definition of addition modulo n.

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Well-definition of multiplication modulo n Prove each of the following important statements about multiplication modulo n, where n is any integer greater than or equal to 2. When applicable, explain how it has changed your life. 1. If a ≡ 0 (mod 7) then 5a ≡ 0 (mod 7) 2. If a ≡ 2 (mod 3)7 then 3a ≡ 6 (mod 3)7. 3. If a ≡ 4 (mod 10), then 12a ≡ 8 (mod 10). 4. If a2 ≡ 4 (mod 7), then a ≡ 2 (mod 7). 5. If a ≡ b (mod n), then 5a ≡ 5b (mod n). 6. If a ≡ b (mod n), then for any integer k, ak ≡ bk (mod n). ∞. If a1 ≡ b1 (mod n) and a2 ≡ b2 (mod n), then a1 a2 ≡ b1 b2 (mod n). The last statement (problem “∞”) is called the well-definition of multiplication modulo n.

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