Show that if a≡b(mod m) and c≡d(mod m), where a, b, c, d ϵ Z with m≥2, then a-c≡b-d(mod m).
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Given an integer , called a modulus, two integers are said to be congruent modulo , if is a divisor of their difference (i.e., if there is an integer such that ). Congruence modulo is denoted: .
If and , then there exist integers and such that and . Then . It follows that where is an integer as a difference of two integers. Therefore,