example
Find the greatest common factor of [latex]24[/latex] and [latex]36[/latex].
Solution
| Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form. |
Factor [latex]24[/latex] and [latex]36[/latex]. |
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| Step 2: List all factors--matching common factors in a column. |
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| In each column, circle the common factors. |
Circle the [latex]2, 2[/latex], and [latex]3[/latex] that are shared by both numbers. |
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| Step 3: Bring down the common factors that all expressions share. |
Bring down the [latex]2, 2, 3[/latex] and then multiply. |
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| Step 4: Multiply the factors. |
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The GCF of [latex]24[/latex] and [latex]36[/latex] is [latex]12[/latex]. |
Notice that since the GCF is a factor of both numbers, [latex]24[/latex] and [latex]36[/latex] can be written as multiples of [latex]12[/latex].
[latex-display]\begin{array}{c}24=12\cdot 2\\ 36=12\cdot 3\end{array}[/latex-display]
example
Find the greatest common factor of [latex]5x\text{ and }15[/latex].
Answer:
Solution
| Factor each number into primes.
Circle the common factors in each column.
Bring down the common factors. |
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The GCF of [latex]5x[/latex] and [latex]15[/latex] is [latex]5[/latex]. |
example
Find the greatest common factor of [latex]12{x}^{2}[/latex] and [latex]18{x}^{3}[/latex].
Answer:
Solution
| Factor each coefficient into primes and write
the variables with exponents in expanded form.
Circle the common factors in each column.
Bring down the common factors.
Multiply the factors. |
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The GCF of [latex]12{x}^{2}[/latex] and [latex]18{x}^{3}[/latex] is [latex]6{x}^{2}[/latex] |
example
Find the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[/latex].
Answer:
Solution
| Factor each coefficient into primes and write
the variables with exponents in expanded form.
Circle the common factors in each column.
Bring down the common factors.
Multiply the factors. |
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The GCF of [latex]14{x}^{3}[/latex] and [latex]8{x}^{2}[/latex] and [latex]10x[/latex] is [latex]2x[/latex] |
In The Language of Algebra we factored numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.
