Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Definition: If a,b,Z and a-0, we say that a divides b if )c,Z(b=ac), denoted by
a | b. When a divides b, we denote a as a factor of b and b as a multiple of a.
When a does not divide b, we denote this as a/|b.
Theorem: Let a,b,c,Z. Then
1. If a | b and a | c, then a | (b+c).
2. If a | b, then a | (bc).
3. If a | b and b | c, then a | c.
Proof: Since a | b, )s,Z(b=as).
1. Since a | c it follows that ) t,Z(c=at). Hence, b+c = as + at = a(s+t).
Therefore, a | (b+c).
2. bc = (as)c = a(sc). Therefore, a | (bc).
3. Since b | c it follows that ) t,Z(c=bt). c = bt = (as)t = a(st), Therefore, a | c.
Corollary: Let a,b,c,Z. If a | b and b | c, then a | (mb+nc) for all m,n,Z.
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Theorem (Division Algorithm): Let a,d,Z(d > 0). Then )!q,r,Z(a = dq+r).
Definition: In the division algorithm, a is the dividend, d is the divisor, q is the
quotient, and r is the remainder. We write q = a div d and r = a mod d.
Examples:
• Consider 101 divided by 9: 101 = 11$9 + 2.
• Consider -11 divided by 3: -11 = 3(-4) + 1.
Definition: Let a,b,m,Z(m > 0). Then a is congruent to b modulo m if m | (a-b),
denoted a ' b (mod m). The set of integers congruent to an integer a modulo m
is called the congruence class of a modulo m.
Theorem: Let a,b,m,Z(m > 0). Then a ' b (mod m) if and only if a mod m = b
mod m.
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