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.

Problem 1

Find the least common multiple of

a) 12 and 16.

b) 39 and 52

c) 24 and 96

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 [pic]5

54 = (18)(3) = (2)(3)(3)(3) = 2[pic]33

24 = (8)(3) = (2)(2)(2)(3) = 23 [pic]3

• The different factors appearing in all the terms are 2, 3, and 5.

maximum of all the 2’s, (2, 23) = 23,

maximum of all the 3’s, (32, 33, 3) = 33,

maximum of all the 5’s, (5) = 5.

• Therefore the LCM = 23[pic] 33[pic]5 = (8)(27)(5) = 1080

Problem 2

a) Find the LCM of 45 and 60.

b) Find the LCM 12, 16, and 48

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 [pic]5 45a2bc5 = 32 [pic]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[pic]5[pic] a2[pic]b2 [pic]c5[pic]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.

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.

[pic], [pic], and [pic]

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:

[pic]

Problem 4

Determine the LCD of the following fractions

a) [pic], [pic] b) [pic], [pic]

c) [pic], [pic], and [pic] d) [pic], [pic], and [pic]

e) [pic], [pic], and [pic] f) [pic], [pic], and [pic]

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The least common multiple is the smallest number in the set of common multiples.

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