Finding
the Least Common Multiples (LCM)
The
least common multiple of a set of numbers is the smallest number
that is a multiple of each number.
To
find the LCM of a set of numbers:
1. Find the prime factorization of each number
in exponential form.
2.
Form a product of all distinct prime factors found in
(1) and raised each common  prime
factor to the highest exponent it is raised to in any of the
factorizations.
Ex
1: Find the LCM of 28 and
42.
Sol:
Find the prime factorization of each number and write it
in exponential form:
The distinct prime factors appear in the factorization
are 2, 3, and 7.
The highest power of 2 is 2, the highest power of 3 is
1, and the highest power of 7 is 1.
Thus LCM
Ex
2:
Find the LCM of 12, 20,
and 54.
Sol:
Find the prime factorization of each number and write it
in exponential form:
Form the product of each distinct prime factor 2, 3, 5,
and raise each prime factor to highest exponent:
Ex
3:
Find the LCM of 12, 18,
24, and 30.
Sol:
Finding the Lowest
Common Denominator (LCD)
The lowest common denominator (LCD) is the smallest number that
is divisible by the denominators of all fractions being considered.
To
find the LCD of two or more fractions, find the least common multiple
(LCM) of their denominators.
Ex
4: Find the
lowest common denominator (LCD) of
.
Sol:
Find the prime factorization of each denominator 28 and
42:
Form the
product of each prime distinct factor 2, 3, 7, and raise each
factor with highest exponent:
LCD  (It is the same
as finding the LCM of 28 and 42.)
Ex
5: Find the lowest
common denominator for fractions
 .
Sol:
Find the prime factorization of each denominator 10, 4,
and 15:
Form the product of each prime distinct factor 2, 3, 5,
and raise each factor with highest exponent:
LCD 
Ex
6:
Add  .
Sol:
LCD = 84
Change
each fraction to an equivalent fraction with the same LCD.
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