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To simpliey a fraction,
a/b, you compute
the gcf or gcd
gcd(a, b) = c
If c = 1 then a/b
is already in lowest terms. Otherwise, the simplified fraction is
then the simplified fraction is
(a/c) / (b/c)
The method taught in most middle schools for computing the gcd is
the tree of factors method. But a more systematic method
is the Eulidean Algorithm which computes remainders
repeatedly to obtain the gcd. For example,
gcd ( 2970, 1265 )
follows this sequence of remainder (r)
computations:
step
abr1 2970 1265 440 2 1265 440 385 3 440 385 55 4 385 55 0
When the remainder becomes zero, the value in colum
b
is the gcd. This algorithm, or step by step procedure, is
credited to Euclid, a Greek mathematician (ca. 300 B.C.).
Even the fastest modern gcd algorithms used on digital computers are
based on the same principle.
Challenge:
Let a and
b be integers, not both zero.
Prove that if any integer
k divides both
a and
b, then
k must divide
gcd(a, b).
The lcm of a and
b, is easy to derive once
gcd(a, b) = c
is computed:
lcm(a, b) = a × (b/c)
Challenge: Figure out how to compute the lcm of several integers
by computing their gcd first.
Use the computation capabilities on the fractions page
to help you experiment with your ideas.
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