Definition 4.1. Let n be a fixed positive integer. Two integers a congruent modulo n, symbolized by
a= b (mod n) if n divides the difference a b; that is, provided that a b = kn for some integer k.
consider n = 7. It is routine to check that 3 = 24 (mod 7)
- 31 = 11 (mod 7) , -15 = -64 (mod 7) because 3 - 24 = (-3)7 -31- 11 = (-6)7, and -15 -(-64) = 7 · 7. When
n A' (a - b), we say that a is incongruent to b modulo n, and in this case we write a ¢ b (mod n). For a simple example: 25 ¢ 12 (mod 7), because 7 fails to divide 25 -12 = 13. It is to be noted that any two integers are congruent modulo .
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