# Greatest Common Factor (GCF) and Least Common multiple (LCM)

The greatest Common factor (GCF) of a set of natural numbers is defined to be the largest natural number that is a factor of each number in the set.

Thus, the GCF of the set of numbers

{8, 12, 16}

Is 4, since 4 is the largest natural number that is a factor, or divisor, of each of the numbers 8, 12, 16.

Example :

Find the gcf of the set {144, 630, 756}.

In this case, the size of the numbers prevents us from determining the gcf simply by inspection. Therefore, we must begin by factoring each of the numbers completely. Using the method described a factor prime, we arrive at the factorizations below :

144 = 2 . 2. 2 . 2. 3 . 3

630 = 2 . 3 . 3. 5 . 7

756 = 2 . 2 . 3 . 3. 3 . 7

Now, since the gcf must be a factor of each of the numbers 144 , 630, and 756, it can contain only the factors that are in common to the three numbers. Moreover, it should contain each common factor as many times as the least number of times it appears in any single factorization. The only factors in common to the three numbers are 2 and 3. The least number of times that 2 appears is once (in 630), so it should appear once in the gcf. The least number of times that 3 appears is twice (in both 144 and 630), so it should appear twice in the gcf. Therefore,

gcf = 2 . 3. 3 = 18

whenever possible, the greatest common factor should be determined simply by inspection. This method, although involving some guesswork, should not be discouraged. Only when it is much too involved to try to guess at the gcf should the method of example be used.

The same can be said about finding the least common multiple of a set of numbers.

The least common multiple (lcm) of a set of natural numbers is the smallest natural number that is a multiple of each number in the set.

Thus, the lcm of the set of numbers

{2, 3} Is 6;

The lcm of the set of numbers {10, 15} is 30; and the lcm of the set of number {4, 6, 12} is 12.

Example :

Find the lcm of the set {24, 90, 75}.

These numbers are too large to determine the lcm by inspection, so we begin by completely factoring each number.

24 = 2. 2. 2. 3

90 = 2 . 3. 3. 5

75 = 3. 5. 5

Now, since the lcm is to be a multiple of 24, 90, and 75, it must include each distinct factor as many times as the most number of times it appears in any single factorization. The most number of times 2 appears is three times (in 24), the most number of times 3 appears is twice (in 90), and the most number of times 5 appears is twice (in 75). Therefore, the lcm must contain three 2’s, two 3’s, and two 5’s.

Lcm = 2 . 2. 2. 3 . 3. 5. 5. = 1800

When working with fractions, it is common practice to refer to the least common multiple (lcm) of a set of denominators as the least common denominator (lcd).