Math 365 Lecture Notes © S. Nite 3/2/2012
Section 5-5
Page 1 of 5
5.5 Greatest Common Divisor and Least Common Multiple
Greatest Common Divisor
The greatest common divisor (GCD) of two natural numbers a and b is the greatest
natural number that divides both a and b.
Example: 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?
Colored Rods Method
The GCD(a, b) is the largest rod that can be used to make one train of length a and
another train of length b.
Example: Find GCD(27, 18).
9-rods, 27 rod
9-rods, 18 rod
Math 365 Lecture Notes © S. Nite 3/2/2012
Section 5-5
Page 2 of 5
Intersection of Sets Method
The GCD(a, b) is the largest element in the intersection of the set of divisors for a
and the set of divisors for b.
Example: Find the GCD(27, 18).
D27 =
D18 =
D27 ∩ D18 =
Prime Factorization Method
The GCD(a, b) is the product of the factors common to a and b.
The prime factorization method is more efficient when the numbers and/or number
of factors is larger.
Example: Find GCD(630, 396).
Two numbers with no common factors are relatively prime.
Example: Which of the following pairs of numbers are relatively prime?
56 and 35
15 and 27
46 and 45
Math 365 Lecture Notes © S. Nite 3/2/2012
Section 5-5
Page 3 of 5
Euclidean Algorithm
The GCD can be found through an iterative process, in which GCD(a, b) is the
divisor that yields a remainder of zero when dividing the divisor by the remainder
repeatedly.
Example: Find GCD(5850, 3300).
Theorem 5-28
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.
Positive
numbers a
and b, a ≠ b
Divide the
larger number
by the smaller.
Is the
remainder
zero?
No
Divide last
divisor by
remainder.
Yes
Last divisor
is the GCD of
a and b.
Math 365 Lecture Notes © S. Nite 3/2/2012
Section 5-5
Page 4 of 5
Least Common Multiple
Suppose a and b are natural numbers. Then the least common multiple (LCM) of a
and b is the least natural number that is simultaneously a multiple of a and a
multiple of b.
Example: Wieners 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 could buy so
that there is an equal number of wieners and buns?
Number-Line Method
Show multiples of the numbers on the number line to see where the multiples match.
Example: Find the LCM(3, 4).
Colored Rods Method
The LCM(a, b) is the smallest rod for which trains of rod length a and rod length b
are the same total length.
Example: Find LCM(4, 6).
4-rods, 6-rods, 10-rod, and 2-rod
Common length for 4-rods and 6-rods is 12.
Math 365 Lecture Notes © S. Nite 3/2/2012
Section 5-5
Page 5 of 5
Intersection of Sets Method
The LCM(a, b) is the smallest element in the intersection of the set of multiples for
a and the set of multiples for b.
Example: Find the LCM(12, 8).
Prime Factorization Method
The LCM(a, b) is the product of all factors in the union of the set of factors of a and
the set of factors of b.
Example: Find LCM(630, 396).
GCD-LCM Product Method
Theorem 5-29
For any two natural numbers a and b, GCD(a, b) LCM(a, b) = ab.
Example: Given GCD(a, b) = 30 and ab = 189000, find the LCM(a, b).
Division by Primes Method
The LCM can be found through an iterative process, in which LCM(a, b) is the
product of prime divisors when a and b and then their divisors are divided by the
common prime numbers repeatedly until a row of ones results.
Example: Find LCM(120, 72, 12).