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2.3. GREATEST COMMON DIVISORS 43
2.3 Greatest Common Divisors
The following definition is valid whenever we have a concept of division.
Definition 25 h is said to be a greatest common divisor, or g.c.d., of f and g
if, and only if:
1. h divides both f and g;
2. if h ′ divides both f and g, then h ′ divides h.
This definition clearly extends to any number of arguments. The g.c.d. is normally
written gcd(f, g).
Note that we have defined a g.c.d, whereas it is more common to talk of the
g.c.d. However, ‘a’ is correct. We normally say that 2 is the g.c.d. of 4 and 6,
but in fact −2 is equally a g.c.d. of 4 and 6.
Proposition 12 If h and h ′ are greatest common divisors of a and b, they are
associates (definition 11).
Greatest common divisors need not exist. For example, let us consider the set
of all integers with √ −5. 2 clearly divides both 6 and and 2 + 2 √ −5. However,
so does 1 + √ −5 (6 = (1 + √ −5)(1 − √ −5)), yet there is no multiple of both 2
and 1 + √ −5 which divides both.
Definition 26 An integral domain (definition 9) in which any two elements
have a greatest common divisor is known 22 as a g.c.d. domain.
If R is a g.c.d. domain, then the elements of the field of fractions (definition 14)
can be simplified by cancelling a g.c.d. between numerator and denominator,
often called “reducing to lowest terms”. While this simplifies fractions, it does
not guarantee that they are normal or canonical. One might think that 0 1
0
was the unique representation of zero required for normality, but what of
−1 ?
Equally −1
2 = 1
−2
, and in general we have to remove the ambiguity caused by
units. In the case of rational numbers, we do this automatically by making the
denominator positive, but the general case is more difficult [DT90].
Definition 27 h is said to be a least common multiple, or l.c.m., of f and g
if, and only if:
1. both f and g divide h ;
2. if both f and g divide h ′ , then h divides h ′ .
This definition clearly extends to any number of arguments. The l.c.m. is
normally written lcm(f, g).
22 Normally known as a unique factorisation domain, but, while the existence of greatest
common divisors is equivalent to the existence of unique factorisation, the ability to compute
greatest common divisors is not equivalent to the ability to compute unique factorisations
[FS56, DGT91], and hence we wish to distinguish the two.