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Unformatted text preview: “An integer a is divisible by b”.
“An integer b divides a”.
What it means?
Deﬁnition. An integer b divides a (b  a) if and
only if there is an integer q such that a = qb,
or equivalently ∃c, a = qb with U of D= Z.
b is called a factor (divisor) of a,
a is called a multiple of b.
b a means NOT ba (b does not divide a),
i.e., ∀c, a = bc 1 Theorem. (Basic properties of division)
(i) If a  b and b  c, then a  c.
(ii) If a  b and a  c, then a  (xb + yc) for any
x, y ∈ Z. In particular, a  (b + c) and a  (b − c).
(iii) If a  b and b  a, then a = ±b.
(iv) If a  b and b = 0, then a ≤ b. 3 Proof.
(i). If b = qa and c = rb for some q, r ∈ Z, then
c = rb = r(qa) = (rq)a.
(ii). If b = qa and c = ra for some q, r ∈ Z,
then xb + yc = x(ba) + y(ra) = (xb)a + (yr)a =
(xb + yr)a.
(iii) If b = qa and a = rb for some q, r ∈ Z, then
a = rb = r(qa) = (rq)a, so that (rq − 1)a = 0.
Case 1. a = 0. Then b = q · 0 = 0, and a = ±b.
Case 2. a = 0. Then rq = 1, so that
r = q = ±1. Hence a = ±b.
(iv) Let b = qa for some q ∈ Z and b = 0.
Then q = 0, so that q ≥ 1.
Hence b = qa ≥ a. 5 Deﬁnition.
Let a and b are integers and b = 0.
If a = qb + r and 0 ≤ r < b, then
q is the quotient, r is remainder,
when a is divided by a.
An integer q is the quotient if and only if qb is
the greatest multiple of b lesser than a.
Examples.
a = 14, b = 3, a = 4b + 2, q = 4, r = 2
a = 14, b = −3, a = −4b + 2, q = −4, r = 2
a = −14, b = 3, a = −5b + 1, q = −5, r = 1
a = −14, b = −3, a = 5b + 1, q = 5, r = 1
7 ...
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 Spring '08
 ANDREWCHILDS

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