Greatest Common Factor and Least Common Multiple

GCF and LCM

Short Cut Methods for GCF

As the number of factors becomes larger, it is not practical to list every factor and search for the greatest. Instead, we can use what we know about prime factorization to help us find the greatest common factor.

Example:   Find GCF (308, 1176).

First, we find the prime factorizations of the numbers.

308 = 2² · 7 · 11 and 1176 = 2³ · 3 · 7²

Next, we examine how the factors correspond. For now, we will expand the exponent notation and line up the factors that match, leaving any other factors at the end of the factorization.

We see that   2 · 2 · 7 are the factors that are in both factorizations.

Therefore the GCF (308, 1176) = 2 · 2 · 7 = 28.

 Further, notice that when the factors are matched up, there are three factors of 2 in 1176 but there are only two factors of 2 in 308. We were only able to match up the lesser number of 2's. This corresponds to the least exponent of the factor 2.

So to get the GCF of two numbers, we only need to take the common prime factors from the prime factorizations and compare their exponents. In each case, we take the exponential expression that has the least exponent and multiply those together.

Example:   We reconsider the previous example to illustrate the use of the exponents.

Find GCF (308, 1176).

First, we find the prime factorizations of the numbers.

  308 = 2² · 7 · 11

1176 = 2³ · 3 · 7²

For the common prime factors, the least exponent for 2 is 2 and the least exponent for 7 is 1. So, in this case we have GCF (308, 1176) = 2² · 7¹ = 4 · 7 = 28.

Example: Find GCF(4950, 7020)

4950 = 2 · 3² · 5² · 11

7020 = 2² · 3³ · 5 · 13

The common prime factors are 2, 3, and 5. The least exponent for 2 and 5 is one (2¹ = 2 and 5¹ = 5) and for 3 is two.

So the GCF (4950, 7020) = 2 · 3² · 5 = 90.

Example: Find GCF(7920, 92664)

7920 = 2⁴ · 3² · 5 · 11

92664 = 2³ · 3⁴ · 11 · 13

The common prime factors are 2, 3, and 11. The least exponent for 2 is 3 (2³), for 3 is 2 (3²), and for 11 is 1 (11¹).

So the GCF (7920, 92664) = 2³ · 3² · 11 = 792.

 

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