Greatest Common Divisor and Lowest Common Multiple
2:48 AM
Muhammad Yusuf
, Posted in
Recall
that the greatest common divisor of two natural numbers a and b is the
largest positive
number that is a factor of both a and b. The
lowest common multiple of a and b is the
smallest positive number that is a multiple of a and b. We
denote the greatest common divisor of a and b by gcd(a; b) and the
lowest common multiple of a and b by lcm(a; b)
REMARK:
Some mathematicians denote the greatest common divisor of a and b just as (a, b).
EXAMPLE
Find the gcd and lcm of the following pairs.
a = 40 and b = 28
Solution:
We can list all of the factors of each
40 : 1, 2, 4, 5, 8, 10, 20, 40
28 : 1, 2, 4, 7, 14, 28
Thus,
gcd(40, 28) = 4.
For
multiples we have
40 : 40, 80, 120, 160, 200, 240, 280, ...
28 : 28, 56, 84, 112, 140, 168, 196, 224, 252, 280, ....
So,
lcm(40, 28) = 280.
EXAMPLE Let a = 784 and
b = 1400. Find gcd(a; b) and lcm(a; b).
Solution: Writing
out the prime factorizations we get
a = 24 x 72 1400 = 23 x 52 x 7
Thus, gcd(784, 1400) = 23 x 7 = 56 and lcm(784, 1400) = 24 x 52 x 72 = 19600.
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