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Number Theory
The Greatest Common Divisor (GCD)
R. Inkulu
http://www.iitg.ac.in/rinkulu/
(GCD)
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Division Algorithm
Given integers a and b, with b > 0, there exists unique integers q (quotient)
and r (remainder) satisfying a = qb + r with 0 ≤ r < b.1
* the existence was proved using well-ordering property earlier
* for the uniqueness part: let a = q0 b + r0 = q00 b + r00 ; then |r0 − r00 | < b and hence |q0 − q00 | < 1
1
extends to b being 6= 0
(GCD)
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Few properties of division
For integers a, b, c,
• a|12 iff a = ±1
• if a|b and b 6= 0, then |a| ≤ |b|
• if a|b and a|c, then a|(bx + cy) for arbitrary integers x and y
2
a divides b with remainder 0 is denoted with a|b; when a|b, we say either of these: a is a
divisor of b, a is a factor of b, or b is a multiple of a
(GCD)
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GCD: definition
Let a and b be integers, with at least one of them different from 0. The
greatest common divisor of a and b, denoted by gcd(a, b), is the positive
integer d satisfying:
• d|a and d|b,
• if c|a and c|b, then c ≤ d.
ex. gcd(8, 17) = 1, gcd(−5, 5) = 5, gcd(−8, −36) = 4
(GCD)
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Elementary properties of gcd
• Let g = gcd(a, b); a = gr, b = gs for integers r and s. Then
gcd(r, s) = 1.
* direct proof of the contrapositive
• Given integers a, b, c, the gcd(a, bc) = 1 iff gcd(a, b) = 1 and
gcd(a, c) = 1.
* ”⇒” proof by contradiction: gcd(a, bc) ≥ gcd(a, b); analogously, gcd(a, bc) ≥ gcd(a, c)
* ”⇐” proof by contradiction: gcd(a, b) (or gcd(a, c)) ≥ prime divisor of gcd(a, bc)
(GCD)
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gcd(a, b) is a linear combination of a and b
For any two integers a and b, not both of which are zero, there exist integers x
and y such that gcd(a, b) = ax + by.
* the smallest element d in the non-empty set
S = {au + bv | au + bv > 0; u, v are integers} is the gcd(a, b)
(GCD)
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Corollaries to above
• If a and b are given integers, not both 0, then the set
T = {ax + by|x, y are integers}
is precisely the set of all multiples of gcd(a, b).
• If a|c and b|c, with gcd(a, b) = 1, then ab|c.
(GCD)
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Relatively prime: definitions
• Two integers a and b, not both of which are 0, are said to be relatively
prime (a.k.a. coprime) whenever gcd(a, b) = 1.
(GCD)
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Relatively prime: definitions
• Two integers a and b, not both of which are 0, are said to be relatively
prime (a.k.a. coprime) whenever gcd(a, b) = 1.
• The integers a1 , a2 , . . . , an are mutually relatively prime if
gcd(a1 , a2 , . . . , an ) = 1.
• The integers a1 , a2 , . . . , an are pairwise relatively prime if, for each pair
of integers ai and aj , gcd(ai , aj ) = 1.
(GCD)
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Relatively prime: definitions
• Two integers a and b, not both of which are 0, are said to be relatively
prime (a.k.a. coprime) whenever gcd(a, b) = 1.
• The integers a1 , a2 , . . . , an are mutually relatively prime if
gcd(a1 , a2 , . . . , an ) = 1.
• The integers a1 , a2 , . . . , an are pairwise relatively prime if, for each pair
of integers ai and aj , gcd(ai , aj ) = 1.
Note that pairwise relatively prime integers must be mutually relatively prime,
but the converse is not necessarily true.
(GCD)
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Few properties related to relative primality
• Let a and b be integers, not both zero. Then a and b are relatively prime
iff there exist integers x and y such that 1 = ax + by.
• Let a and b be integers with gcd(a, b) = d; then gcd( da , db ) = 1.
(GCD)
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Euclid’s lemma
If a|bc, with gcd(a, b) = 1, then a|c.
(GCD)
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Yet another view of GCD
Let a, b be positive integers. For a positive integer d, d = gcd(a, b) iff
(i) d|a and d|b,
(ii) whenever c|a and c|b, then c|d.
(GCD)
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LCM: definition
The least common multiple of two nonzero integers a and b, denoted by
lcm(a, b), is the positive integer m satisfying
• a|m and b|m,
• if a|c and b|c, with c > 0, then m ≤ c.
(GCD)
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Relating gcd and lcm
For any two positive integers a and b, gcd(a, b) ∗ lcm(a, b) = a ∗ b
* m=
ab
gcd(a,b)
(GCD)
divides any common multiple c of a and b, hence, m is the lcm(a, b)
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Two corollaries
• Every common multiple of positive integers a and b is divisible by
lcm(a, b).
• For any choice of positive integers a and b, lcm(a, b) = ab iff
gcd(a, b) = 1.
(GCD)
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