Greatest Common Factor and Least Common Multiple

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There are a variety of techniques for finding the LCM and GCF. The two most common strategies involve making a list, or using the prime factorization.

For example, the LCM of 5 and 6 can be found by simply listing the multiples of \(5\) and \(6\), and then identifying the lowest multiple shared by both numbers.

\(5, 10, 15, 20, 25, \mathbf{30}, 35…\)
\(6, 12, 18, 24, \mathbf{30}, 36…\)
\(\mathbf{30}\) is the LCM.

Similarly, the GCF can be found by listing the factors of each number, and then identifying the greatest factor that is shared. For example, the GCF of \(40\) and \(32\) can be found by listing the factors of each number.

\(40\): \(1, 2, 4, 5, \mathbf8, 10, 20, 40\)
\(32\): \(1, 2, 4, \mathbf8, 16, 32\)
\(\mathbf8\) is the GCF.

For larger numbers, it will not be realistic to make a list of factors or multiples to identify the GCF or LCM. For large numbers, it is most efficient to use the prime factorization technique.

For example, when finding the LCM, start by finding the prime factorization of each number (this can be done by creating a factor tree). The prime factorization of \(20\) is \(2\times2\times5\), and the prime factorization of \(32\) is \(2\times2\times2\times2\times2\). Circle the factors that are in common and only count these once.

Now multiply all of the factors (remember not to double-count those circled \(2\)s). This becomes \(2\times2\times5\times2\times2\times2\), which equals \(160\). The LCM of \(20\) and \(32\) is \(160\).

When finding the GCF, start by listing the prime factorization of each number (this can be done by creating a factor tree). For example, the prime factorization of \(45\) is \(5\times3\times3\), and the prime factorization of \(120\) is \(5\times3\times2\times2\times2\). Now simply multiply all of the factors that are shared by both numbers. In this case, we would multiply \(5\times3\) which equals \(15\). The GCF of \(45\) and \(120\) is \(15\).

The prime factorization approach can seem like a fairly lengthy process, but when working with large numbers it is guaranteed to be a time-saver.

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There are two main strategies for finding the GCF: Listing the factors, or using the prime factorization.

The first strategy involves simply listing the factors of each number, and then looking for the greatest factor that is shared by both numbers. For example, if we are looking for the GCF of \(36\) and \(45\), we can list the factors of both numbers and identify the largest number in common.

\(36\): \(1,2,3,4,6,\mathbf9,12,18,36\)
\(45\): \(1,3,5,\mathbf9,15,45\)
The GCF of \(36\) and \(45\) is \(\mathbf9\).

Listing the factors of each number and then identifying the largest factor in common works well for small numbers. However, when finding the GCF of very large numbers it is more efficient to use the prime factorization approach.

For example, when finding the GCF of \(180\) and \(162\), we start by listing the prime factorization of each number (this can be done by creating a factor tree). The prime factorization of \(180\) is \(2\times2\times3\times3\times5\), and the prime factorization of \(162\) is \(2\times3\times3\times3\times3\). Now look for the factors that are shared by both numbers. In this case, both numbers share one \(2\), and two \(3\)s, or \(2\times3\times3\). The result of \(2\times3\times3\) is \(18\), which is the GCF! This strategy is often more efficient when finding the GCF of really large numbers.

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The GCF stands for the “greatest common factor”. The GCF is defined as the largest number that is a factor of two or more numbers. For example, the GCF of \(24\) and \(36\) is \(12\), because the largest factor that is shared by \(24\) and \(36\) is \(12\). \(24\) and \(36\) have other factors in common, but \(12\) is the largest.

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There are a variety of techniques for finding the lowest common multiple. Two common approaches are listing the multiples, and using the prime factorization. Listing the multiples is just as it sounds, simply list the multiples of each number, and then look for the lowest multiple shared by both numbers. For example, when finding the lowest common multiple of \(3\) and \(4\), list the multiples:

\(3\): \(3,6,9,\mathbf{12},15,18…\)
\(4\): \(4,8,\mathbf{12},16,20…\)
\(\mathbf{12}\) is the lowest multiple shared by \(3\) and \(4\).

Listing the multiples is a great strategy when the numbers are fairly small. When numbers are large, such as \(38\) and \(42\), we should use the prime factorization approach. Start by listing the prime factorization of each number (this can be done using a factor tree).

\(38\): \(2\times19\)
\(42\): \(2\times3\times7\)

Now circle the shared factors (only count these once).

Now multiply all of the factors (remember to only count the \(2\)s once). This becomes \(2\times19\times3\times7\), which equals \(798\). The LCM of \(38\) and \(42\) is \(798\).

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Taking out the LCM is a helpful skill when adding or subtracting fractions. Determining the lowest common multiple creates a denominator that is the same for both fractions. For example, the common denominator for \(\frac{2}{7}+\frac{3}{5}\) would be \(35\), because \(35\) is the LCM of \(7\) and \(5\). The new fractions become \(\frac{10}{35}+\frac{21}{35}\), which equals \(\frac{31}{35}\).