Basic Mathematics for College Students

The greatest exponent on the factor 2 is 3.

The greatest exponent on the factor 3 is 1.

The greatest exponent on the factor 5 is 1.

The LCM of 24 and 60 is

23  31  51  8  3  5  120 Evaluate: 2^3 8.

60  22  31  51

24  23  31

94 Chapter 1 Whole Numbers

EXAMPLE (^5) Find the LCM of and 45.
StrategyWe will begin by finding the prime factorizations of and 45.
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 and 45.
This can be written as
This can be written as.
This can be written as.
Step 2The prime factorizations of and 45 contain the prime factors
and 7. 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 two times, because 2 appears two times in the
  factorization of 28. Circle as shown above.


• We will use the factor 3 twice, because it appears two times in the
  factorization of 45. Circle as shown above.


• We will use the factor 5 once, because it appears one time in the
  factorization of 45. Circle the 5, as shown above.


• We will use the factor 7 once, because it appears one time in the
  factorization of 28 and one time in the factorization of 42. You may circle
  either 7, but only circle one of them.
  Since there are no other prime factors in either prime factorization, we have
  Use the factor 2 two times.
  Use the factor 3 two times.
  Use the factor 5 one time.
  Use the factor 7 one time.

If we use exponents, we have

Either way, we have found that the LCM Note that 1,260 is

the smallest number that is divisible by and 45:

1,260

45

 28

1,260

42

 30

1,260

4

 315

28, 42,

(28, 42, 45)1,260.

LCM (28, 42, 45) 22  32  5  7 1,260

LCM (28, 42, 45) 2  2  3  3  5  7 1,260

   

3 3,

2 2,

(28, 42, 45),

28, 42, 2, 3, 5,

45  3  3  5 32  5

42  2  3  7 21  31  71

28  2  2  7 22  71.

28, 42,

28, 42,

28, 42,

Self Check 5

Find the LCM of

and 75.

Now TryProblem 45

45, 60,

EXAMPLE (^6) Patient Recovery Two patients recovering from heart
surgery exercise daily by walking around a track. One patient can complete a lap
in 4 minutes. The other can complete a lap in 6 minutes. If they begin at the same
time and at the same place on the track, in how many minutes will they arrive
together at the starting point of their workout?

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