5-5 Greatest Common Divisor and Least Common Multiple
Methods to Find the Greatest Common Divisor
Methods to Find the Least Common Multiple
Greatest Common Divisor
Two bands are to be combined to march in a parade. A 24-member band will march behind a
30-member band. The combined bands must have the same number of columns. Each column
must be the same size. What is the greatest number of columns in which they can march?
The bands could each march in 2 columns, and we would have the same number of columns,
but this does not satisfy the condition of having the greatest number of columns.
The number of columns must divide both 24 and 30.
Numbers that divide both 24 and 30 are 1, 2, 3, and 6. The greatest of these numbers is 6.
The first band would have 6 columns with 4 members in each column, and the second band
would have 6 columns with 5 members in each column.
Greatest Common Divisor (GCD)
The greatest common divisor (GCD) of two natural numbers a and b is the greatest natural
number that divides both a and b.
Colored Rods Method
Find the GCD of 6 and 8 using the 6 rod and the 8
rod.
Find the longest rod such that we can use multiples of that rod to build both the 6 rod and the 8
rod.
The 2 rods can be used to build both the 6 and 8 rods.
The 3 rods can be used to build the 6 rod but not the 8 rod.
The 4 rods can be used to build the 8 rod but not the 6 rod.
The 5 rods can be used to build neither.
The 6 rods cannot be used to build the 8 rod.
Therefore, GCD(6, 8) = 2.
The Intersection-of-Sets
Sets Method
List all members of the set of positive divisors of the two integers, then find the set of all
common divisors, and, finally, pick the greatest element in that set.
To find the GCD of 20 and 32, denote the sets of divisors of 20 and 32 by D20 and D32,
respectively.
Because the greatest number in the set of common positive divisors is 4, GCD(20, 32) = 4.
The Prime Factorization Method
To find the GCD of two or more positive integers, first find the prime factorizations of the given
numbers and then identify each common
common prime factor of the given numbers. The GCD is the
product of the common factors, each raised to the lowest power of that prime that occurs in
any of the prime factorizations.
Numbers, such as 4 and 9, whose GCD is 1 are relatively prime.
Example 1 Find each of the following:
a. GCD(108, 72)
b. GCD(0, 13)
c. GCD(x, y) if x = 23 · 72 · 11 · 13 and y = 2 · 73 · 13 · 17
d. GCD(x, y, z) if x = 23 · 72 · 11 · 13, y = 2 · 73 · 13 · 17, and z = 22 · 7
e. GCD(x, y) if x = 54 · 1310 and y = 310 · 1120
Calculator Method
Try finding the Greatest Common Divisor on a calculator.
Euclidean Algorithm Method
If a and b are any whole numbers greater than 0 and a ≥ b, then GCD(a, b) = GCD(r, b), where r
is the remainder when a is divided by b.
Finding the GCD of two numbers by repeatedly using the theorem above until the remainder is
0 is called the Euclidean algorithm.
Example 2 Use the Euclidean algorithm to find GCD(10764, 2300).
Example 3 a. Find GCD(134791, 6341, 6339).
b. Find the GCD of any two consecutive whole numbers.
Least Common Multiple
Hot dogs are usually sold 10 to a package, while hot dog buns are usually sold 8 to a package.
What is the least number of packages of each you must buy so that there is an equal number of
hot dogs and buns?
Least Common Multiple (LCM)
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.
Number-Line Method
Find LCM(3, 4).
Beginning at 0, the arrows do not coincide until the point 12 on the number line. Thus, 12 is
LCM(3, 4).
Colored Rods Method
Find LCM(3, 4) using the 3 rod and the 4 rod.
Build trains of 3 rods and 4 rods until they are the same length. The LCM is the common length
of the train.
The Intersection-of-Sets Method
List all members of the set of positive multiples of the two integers, then find the set of all
common multiples, and, finally, pick the least element in that set.
To find the LCM of 8 and 12, denote the sets of positive multiple of 8 and 12 by M8 and M12,
respectively.
Because the least number in the set of common positive multiples is 24, LCM(8, 12) = 24.
Because the least number in the set of common positive multiples is 24, LCM(8, 12) = 24.
The Prime Factorization Method
To find the LCM of two
o natural numbers, first find the prime factorization of each number. Then
take each of the primes that are factors of either of the given numbers. The LCM is the product
of these primes, each raised to the greatest power of the prime that occurs in either of the
prime factorizations.
Example 4 Find LCM(2520, 10530).
GCD-LCM Product Method
For any two natural numbers a and b, GCD(a, b) · LCM(a, b) = ab.
Example 5 Find LCM(731, 952).
Division-by-Primes Method
To find LCM(12, 75, 120), start with the least prime that divides at least one of the given
numbers. Divide as follows:
Because 2 does not divide 75, simply bring down the 75. To obtain the LCM using this
procedure, continue the division process until the row of answers consists of relatively prime
p
numbers as shown next.
5-5
5 Homework # 1, 2, 3, 4, 5, 7, 8, 9, 10, 14,1 5, 16, 17, 18, B-3, 4