1

Slide 4- 1Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

4-5 Greatest Common Divisor and Least Common Multiple

DefinitionThe greatest common divisor (GCD) of two integers a and bis the greatest integer that divides both a and b.

The largest rod that we can use multiples of to build both the 12 and 16 rod is 4. So the GCD(12, 16) = 4.

Colored Rods Method

Slide 4- 2Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Intersection on-of-Sets Method

Finding the Greatest Common Divisor

First, find the sets of all possible divisors of 12 and 16.

The set of common positive divisors of 12 and 16 is:

Because 4 is the greatest number in this set of common divisors, the GCD of 12 and 16, written GCD(12, 16) = 4.

Slide 4- 3Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Prime Factorization Method

Finding the Greatest Common Divisor

Find the greatest common divisor of 140 and 120.

The prime factorizations are:

The most factors in common are:

GCD(140,120) = 2×2 ×5 = 20

Slide 4- 4Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Euclidean Algorithm MethodFinding the Greatest Common Divisor

Find the GCD of 1492 and 1008.

Slide 4- 5Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Euclidean Algorithm MethodFinding the Greatest Common Divisor

Find the GCD of 1492 and 1008.

GCD(1492, 1008) = GCD(1008, 484)

GCD(1008, 484) = (484, 40)

GCD(484, 40) = GCD(40, 4)

GCD(40, 4) = GCD(4, 0) = 4

Slide 4- 6Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Finding the Greatest Common Divisor

Theorem 4-7

2

Slide 4- 7Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Definition

Least Common Multiple

Suppose that a and b are positive integers. Then the least common multiple (LCM) of a and b is the least positive integer that is simultaneously a multiple of a and a multiple of b.

Find the Least Common Multiple of 6 and 8.

We are looking for the least number that is a multiple of 6 and a multiple of 8. Stated another way, we are looking for the smallest positive integer divisible by both 6 and 8.

Slide 4- 8Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Colored Rods Method

Least Common Multiple

We build trains of 6 rods and 8 rods until they are the same length.

Least common multiple of 6 and 8 is 24

Slide 4- 9Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Intersection-of-Sets Method

Least Common Multiple

Find the least common multiple of 12 and 16.

First, find the set of all positive multiples of both numbers:

The set of common multiples is:

Since the least of these elements is 48, the least common multiple of 12 and 16 is 48, written LCM(12, 16) = 48.

Slide 4- 10Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Intersection-of-Sets Method

Least Common Multiple

Find the least common multiple of 12 and 16.

First, find the set of all positive multiples of both numbers:

The set of common multiples is:

Since the least of these elements is 48, the least common multiple of 12 and 16 is 48, written LCM(12, 16) = 48.

Slide 4- 11Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Prime Factorization MethodLeast Common Multiple

Find the least common multiple of 12 and 16.

Find the prime factorization of each number:

Take each of the primes that are factors of either number, which in this case would be 2 and 3.

Raise each of these primes to the greatest power of the prime that occurs in either of the prime factorizations. The product is the least common multiple.

Slide 4- 12Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Relationship Between GCD and LCM

Given 12 and 15, first find the prime factorization of both numbers:

Notice that 12 • 16 = GCD(12, 16) • LCM(12, 16)

= 4 • 48

= 192

3

Slide 4- 13Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Relationship Between GCD and LCM

For any two natural numbers a and b,

GCD(a,b) • LCM(a,b) = ab.

Theorem 4-8

Suppose we wish to find the least common multiple of 640 and 1296.Applying the Euclidean Algorithm, we can determine that GCD(640, 1296) = 16.

From Theorem 4-8, we know that

16 • LCM(640, 1298) = 640 • 1296

LCM(640, 1296) = (640 • 1296) / 16 = 51,840