1.8 The Least Common Multiple and the Greatest Common Factor 93Finding the LCM Using Prime FactorizationTo find the least common multiple of two (or more) whole numbers:- Prime factor each number.
- The LCM is a product of prime factors, where each factor is used the
greatest number of times it appears in any one factorization.
EXAMPLE (^4) Find the LCM of 24 and 60.
StrategyWe will begin by finding the prime factorizations of 24 and 60.
WHYTo find the LCM, we need to determine the greatest number of times each
prime factor appears in any one factorization.
Solution
Step 1Prime factor 24 and 60.
Division ladders (or factor trees) can be
used to find the prime factorizations.
Step 2The prime factorizations of 24 and 60 contain the prime factors and 5.
To find the LCM, we use each of these factors the greatest number of times it
appears in any one factorization.
- We will use the factor 2 three times, because 2 appears three times in the
factorization of 24. Circle as shown below. - We will use the factor 3 once, because it appears one time in the
factorization of 24 and one time in the factorization of 60. When the
number of times a factor appears are equal, circle either one, but not both,
as shown below. - We will use the factor 5 once, because it appears one time in the
factorization of 60. Circle the 5, as shown below.
Since there are no other prime factors in either prime factorization, we have
Use 2 three times.
Use 3 one time.
Use 5 one time.Note that 120 is the smallest number that is divisible by both 24 and 60:
and120
60
2
120
24
5
LCM (24, 60) 2 2 2 3 5 120
60 2 2 3 5
24 2 2 2 3
2 2 2,
2, 3,
60 2 2 3 5
24 2 2 2 3
Self Check 4
Find the LCM of 18 and 32.
Now TryProblem 37In Example 4, we can express the prime factorizations of 24 and 60 using
exponents. To determine the greatest number of times each factor appears in any
one factorization, we circle the factor with the greatest exponent.
224
212
26
3260
230
315
5