Alternate Methods for Finding LCM and GCFAuthor(s): George McCabe Jr.Source: The Mathematics Teacher, Vol. 72, No. 1 (JANUARY 1979), pp. 34-35Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27961507 .

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sharing teaching ideas

Alternate Methods for Finding LCM and GCF

It is no secret that many youngsters have a great deal of difficulty reducing fractions to lowest terms or determining a lowest common denominator. I am going to pre sent two somewhat different methods for

finding the least common multiple (LCM) and the greatest common factor (GCF) of two numbers. These techniques are some

what limited because they do deal with only two numbers at a time. They do, however, offer some advantages over the conven

tional methods. The procedure for finding the LCM is to

divide the larger of two numbers by their

greatest common factor and then multiply this result by the smaller of the two num

bers. The subtraction feature in this process makes the task easy by decreasing the size

of one of the numbers, thus making the GCF evident.

1. Subtract the smaller number from the

larger (see fig. 1).

Wmi??:

'i ? - .'h. wl"^^n-:v;v^v)-ViiVi?;yl| ?iMiii'Tiir-v; y; "i -y,

Fig. 1. Finding the LCM of 12 and 16

2. Place the result from step 1 over the

larger of the two numbers in fraction form.

3. Reduce this resulting fraction if pos sible.

4. Multiply the denominator of this re

duced fraction by the smaller of the

original two numbers. This is the

LCM for the two numbers.

My alternate method for finding the

GCF first determines if a common factor

exists and then calculates it. The technique was developed to check on step 3 of the first

process or for use by itself. It eliminates all

guesswork from fraction reduction and

does not require memorization of primes. The procedure involves subtracting num

bers with common factors until the remain

der is 0 or 1. The procedure is essentially the Euclidean algorithm, which is studied

in higher mathematics.

1. Divide the larger number by the

smaller number (fig. 2).

Fig. 2. Finding the GCF of 12 and 16

Sharing Teaching Ideas offers practical tips on the teaching of topics related to the secondary

school curriculum. We hope to include classroom-tested approaches that offer new slants on

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34 Mathematics Teacher

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2. If the remainder is 1 or 0, go to step 6.

3. If the remainder is not 1 or 0, divide the remainder from step 1 into the divisor in step 1.

4. If the remainder from step 3 is 1 or 0,

go to step 6.

5. If the remainder from step 3 is not 1 or

0, repeat steps 3 and 4 until the re

mainder is 1 or 0. At that time, step 6 will apply (fig. 3).

6. If the remainder is zero, the divisor is

the GCF. If the remainder is 1, there is no common factor larger than 1.

Several teachers have found that these methods are quite easy for youngsters to

apply. Guesswork based on memorization of primes is eliminated, and the second

process can be used to check on the first. The first process has resulted in students

being able to determine the LCM even

without the use of paper and pencil.

Fig. 3. Finding the GCF of 15 and

George McCabe, Jr. Glen Rock High School Glen Rock, NJ 07452

Your Word Problems:

They'll Look Forward to Them For a first-year high school student,

nothing surpasses the joy of forthcoming word problems, except first-period physical education (swimming for the girls) or a

10:15 a.m. lunch period. If you use an irra tional imagination and often unreal num

bers, however, traditional word problems (mixture, distance, age, and so on) can be come fun instead of a chore. Using my athletic ability as a track coach (zero is a real number) and my humility (ex ponential) as examples, my students plunge into word problems. Characters from sto ries read in English classes (see problem 6), as well as television personalities, are used.

Feel free to use any of these problems, substituting your name in the following ex

amples. Be sure to tell your students not to look for your name in the Guinness Book of

Records. That would be cheating.

1. At 3:00 p.m. Mario and Retti left Hoffman Estates High School to drive their

son Ever home from school. At 3:30 p.m.

Mr. Alex realized that Ever had not lived

up to his name, for he had forgotten to take his algebra book home .(a sin); so Mr. Alex ran after the car and caught it in one min ute. If Flash Alex's battery allowed him to

average 200 kph faster than Mario and

Ever, find Alex's rate.

2. Dr. and Mrs. Duck had three children in consecutive years: Disco, Aqua, and Via. In five years, five times Disco's age (the oldest) will be 116 more than twice the sum

of Aqua's and Via's (the youngest) ages. How old will Disco be next year?

3. One day the Lone Ranger decided he needed additional silver bullets; so he sent

Tonto north to the mine. As Tonto was

leaving, the Ranger went south to K-Mart to see his girlfriend. It took Tonto one hour

more than twice as long to reach the mine as it took the Ranger to reach K-Mart. It's

750 miles from the mine to K-Mart. If the

January 1979 35

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