Basic Mathematics for College Students

1.8 The Least Common Multiple and the Greatest Common Factor 95

3 Find the GCF by listing factors.

We have seen that two whole numbers can have common multiples. They can also
have common factors.To explore this concept, let’s find the factors of 26 and 39 and
see what factors they have in common.
To find the factors of 26, we find all the pairs of whole numbers whose product is

26. There are two possibilities:

Each of the numbers in the pairs is a factor of 26. From least to greatest, the factors
of 26 are and 26.
To find the factors of 39, we find all the pairs of whole numbers whose product is

39. There are two possibilities:

Each of the numbers in the pairs is a factor of 39. From least to greatest, the factors of
39 are and 39. As shown below, the common factorsof 26 and 39 are 1 and 13.

These are the factors of 26.
These are the factors of 39.
Because 13 is the largest number that is a factor of both 26 and 39, it is called the
greatest common factor (GCF)of 26 and 39. We can write this in compact form as:

GCF (26, 39) 13 Read as “The greatest common factor of 26 and 39 is 13.”

1 , 3 , 13 , 39

1 , 2 , 13 , 26

1, 3, 13,

1  39  39 3  13  39

1, 2, 13,

1  26  26 2  13  26

The Greatest Common Factor (GCF)

The greatest common factorof two whole numbers is the largest common

factor of the numbers.

Self Check 6

AQUARIUMS A pet store owner

changes the water in a fish

aquarium every 45 days and he

changes the pump filter every

20 days. If the water and filter are

changed on the same day, in how

many days will they be changed

again together?

Now TryProblem 87

StrategyWe will find the LCM of 4 and 6.

WHYSince one patient reaches the starting point of the workout every 4
minutes, and the other is there every 6 minutes, we want to find the least common
multiple of those numbers. At that time, they will both be at the starting point of
the workout.

Solution
To find the LCM, we prime factor 4 and 6, and circle each prime factor the
greatest number of times it appears in any one factorization.

Use the factor 2 two times, because 2 appears

two times in the factorization of 4.

Use the factor 3 once, because it appears one

time in the factorization of 6.

Since there are no other prime factors in either prime factorization, we have

The patients will arrive together at the starting point 12 minutes after beginning
their workout.

LCM (4, 6) 2  2  3  12

6  2  3

4  2  2

EXAMPLE (^7) Find the GCF of 18 and 45.
StrategyWe will find the factors of 18 and 45.
WHYThen we can identify the largest factor that 18 and 45 have in common.
Self Check 7
Find the GCF of 30 and 42.
Now TryProblem 49