Download Worksheet # The least Common Multiple (LCM)

Worksheet #
The least Common Multiple (LCM)___
Name___________________
The Least Common Multiple (L.C.M.) of a set of numbers is the smallest number that each of the
numbers in the given set can divide into it with a remainder of zero. To add or subtract fractions
with different denominators, the least common multiple of the denominators is most convenient
denominator to use.
One method of finding the least common multiple is to write down the multiples of the given
numbers in a list. Then find what numbers are shared by the numbers in the lists, and pick the
smallest. Using this method, we can define the least common multiple as the smallest number in
the set of common multiples.
The least common multiple of a and b is the least nonzero number that is a common multiple of
a and b.
Example 1
Find the common multiple of 24 and 18.
Solution:
Multiples of 24: { 24, 48, 72, 96, 120, 144, 168, 192, 216, 240 …}
Multiples of 18: { 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, …}
Common Multiples of 12 and 18: {72, 144, …}
The least common multiple of 24 and 18 is 72.
The least common
multiple is the
smallest number in
the set of common
multiples.
Problem 1
Find the least common multiple of
a) 12 and 16.
b) 39 and 52
c) 24 and 96
©2001 Michael Aryee
Least Common Multiple
Page 1
Another method of finding the least common multiple is to first write each number as a product
its prime numbers. Then multiply together all the prime factors, using the common factors only
once and taking only the highest exponent of each prime factor. In this case you must express
each number or expression in an exponential form and select the factors with the largest
exponent without repeating any of the factors.
Example 2
Find the Least Common Multiple of 45, 54, and 24.
Solution:



Write each number as a product of primes.
45 = (9)(5) = (3)(3)(5)
= 32  5
54 = (18)(3) = (2)(3)(3)(3)
= 2  33
24 = (8)(3) = (2)(2)(2)(3)
= 23  3
The different factors appearing in all the terms are 2, 3, and 5.
maximum of all the 2’s,
(2, 23)
maximum of all the 3’s,
(32, 33, 3) = 33,
maximum of all the 5’s,
(5)
= 23,
= 5.
Therefore the LCM = 23  33  5 = (8)(27)(5) = 1080
Problem 2
a) Find the LCM of 45 and 60.
b) Find the LCM 12, 16, and 48
©2001 Michael Aryee
Least Common Multiple
Page 2
Example 3
Find the Least Common Multiple of 45a2bc5 and 27ab2c3d.
Solution:

Write each number as a product of primes.
45 = (9)(5) = (3)(3)(5) = 32  5
45a2bc5 = 32  5 a2 b c5
we have,
27 = (9)(3) = (3)(3)(3) = 33


27ab2c3d = 33 ab2c3d
The different factors appearing in all the terms are 3, 5, a, b, c, and d.
maximum of all the 3’s, (32, 33)
= 33,
maximum of all the 5’s, (5)
= 5,
maximum of all the a’s, (a2, a)
= a2,
maximum of all the b’s, (b, b2)
= b2,
maximum of all the c’s, (c5, c3)
= c5,
maximum of all the d’s, (d)
= d.
Therefore the LCM = 33  5  a2  b2  c5  d = (27)(5)a2b2c = 135a2b2c5d
Problem 3
a)
Find the LCM of 8p4t3 and 36p2t5r.
b) Find the LCM of 2x2y3z, 6xy2z2, and 4x3y3z3.
c) Find the LCM of 5(x – 2)2, 2(x – 2)3, and 10(x – 2).
d) Find the LCM of 12(x – 1)3(y + 1)6z6 , 6(x – 1)4(y + 1)4z4, and 4(x – 1)2(y + 1)5z3.
©2001 Michael Aryee
Least Common Multiple
Page 3
The Least Common Denominator (LCD)
The least common denominator of several fractions or rational expressions is the LCM of the
individual denominators. LCD is the smallest number that is evenly divisible by all of the
denominators of the given fractions. To add or subtract fractions with different denominators, the
least common multiple of the denominators is most convenient denominator to use.
Example 4
Determine the LCD of the following fraction and then determine an equivalent fraction with the
LCD as the new denominator.
7
3
2
,
, and
2
15 x 35 y
9x 2 y
Solution: the denominators are 15x, 35y2, and 9x2y.
Write each number as a product of primes. 15 = (3)(5), 35 = (5)(7), and 9 = (3)(3) = 32
The different factors appearing in all the terms are 3, 5, 7, x, and y.
maximum of all the 3’s,
(3, 32)
= 32,
maximum of all the 5’s,
(5, 5)
= 5,
maximum of all the 7’s,
(7)
= 7,
maximum of all the x’s,
(x, x2)
= x2,
maximum of all the y’s,
(y2, y)
= y2.
LCD = (32)(5)(7)( x3)( y2) = .
Equivalent fractions:
7  21xy 2
63xy 2

,
15 x  21xy 2
315 x 2 y 2
©2001 Michael Aryee
3  9x 2
18 x 2

,
35 y 2  9 x 2 315 x 2 y 2
Least Common Multiple
2  35 y
70 y

2
9 x y  35 y 315 x 2 y 2
Page 4
Problem 4
Determine the LCD of the following fractions
a)
1 3
,
x 2y
b)
2 5
,
a b2
c)
2
1
2
,
, and 2
3
2 2
5u 15u v
3v
d)
9
7
2
, 2 , and
2
xyz
xy z x yz
e)
1
2
3
,
, and
2
2
( x  1) ( x  2)
( x  2) ( x  1)
f)
2
1
2
, 2
, and 2
x
x( x  1) x ( x  1)
©2001 Michael Aryee
Least Common Multiple
Page 5