Looking through the Lens

What's behind the lens?
OBJECTIVE: To learn how to find the greatest common factor (GCF)
[br]In previous lessons, you've learned about prime factorization, prime and composite numbers, factors and multiples, and divisibility rules.[br][br][i][color=#0000ff]Climbing Down Trees:[/color][/i] https://www.geogebra.org/m/hxmuf2dg[br][i][color=#0000ff]Going Dutch:[/color][/i] https://www.geogebra.org/m/xvdee7pg[br][i][color=#0000ff]Sifting Grains:[/color][/i] https://www.geogebra.org/m/gamspcmt[br][br]In this lesson, we'll put these concepts together to find the [color=#0000ff]GREATEST COMMON FACTOR (GCF)[/color], also known as [color=#0000ff]GREATEST COMMON DIVISOR (GCD)[/color] or [color=#0000ff]HIGHEST COMMON FACTOR (HCF).[br][br][/color]The [color=#0000ff]GREATEST COMMON FACTOR (GCF)[/color] of two or more counting numbers is the greatest counting number that evenly divides all the given numbers. If that number is 1, then the two numbers are said to be [color=#0000ff]RELATIVELY PRIME.[/color] There are many approaches to finding the [color=#0000ff]GREATEST COMMON FACTOR[/color] or [color=#0000ff]GCF[/color]. The approach we'll use here will involve [color=#0000ff]PRIME FACTORIZATION[/color] of the numbers whose [color=#0000ff]GCF[/color] is being sought. The applet found in the lesson [color=#0000ff]Climbing Down Trees [/color]can be used here. The prime factors obtained there can be listed for all the numbers and the common factors identified. The number obtained by multiplying the common factors is the [color=#0000ff]GCF[/color].[br][br]Example: What is the GCF of 16 and 24?[color=#0000ff][br][br]Prime factors of 16 : [/color][color=#ff0000]2 x 2 x 2[/color][color=#0000ff] x 2[br]Prime factors of 24 : [/color][color=#ff0000]2 x 2 x 2[/color][color=#0000ff] x 3[br][br][/color]The [color=#0000ff]product[/color] of the common factors, [color=#ff0000]2 x 2 x 2[/color] (shown in [color=#ff0000]red[/color]) is the [color=#0000ff]GCF[/color], which is [color=#ff0000]8[/color].
VENN DIAGRAM
[br]A [color=#0000ff]Venn Diagram[/color] can be used to visualize the concept of the [color=#0000ff]GCF[/color]. Using the example above, we can list the common prime factors of 16 and 24 inside the [color=#0000ff]lens[/color] (the [color=#0000ff]intersection[/color] of the two sets). The other prime factors are listed inside their respective [color=#0000ff]crescents[/color]. We now multiply the numbers inside the lens to find the [color=#0000ff]GCF[/color], which is [color=#ff0000]8[/color].
A more compact method of finding the GCF is through [color=#0000ff]CONTINUOUS DIVISION[/color] by prime factors, as illustrated below.[br][br][img]data:image/png;base64,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[/img][br]Start by dividing by the smallest common prime factor, and repeat the process until no common prime factor can be found. Multiply the common prime factors at the left side to find the [color=#0000ff]GCF[/color].[br][br]The [color=#0000ff]GCF[/color] is 2 x 2 x 2 or [color=#ff0000]8[/color].[br][br]Here's an example using three numbers.[br][br][img]data:image/png;base64,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[/img][br]The [color=#0000ff]GCF[/color] is 2 x 2 x 3 x 3 or [color=#ff0000]36[/color].[br][br]The process just discussed is the systematic way of finding the GCF. For bigger numbers, an [color=#0000ff]obviously[/color] common factor, [color=#ff7700]NOT NECESSARILY PRIME[/color], can be factored out first before the systematic process is applied. Consider this.[br][br][img]data:image/png;base64,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[/img][br]The [color=#0000ff]GCF[/color] is 10 x 2 x 3 x 7 or [color=#ff0000]420[/color].
Use the applet below for practice.[br][br]Enter 2 numbers in the entry boxes. Try solving the problem first on a separate sheet of paper, and then verify your answer by clicking [color=#6aa84f]Solutions[/color]. The GCF will appear below the numbers.[br][br]Click [color=#6aa84f]G.C.F. for 3 Numbers[/color] to work on 3 numbers, and follow the same procedure.[br][br]Repeat as many times as needed to master the concept.
In this lesson, you learned how to find the GREATEST COMMON FACTOR.
In the next lesson, you're going to learn how to find the LEAST COMMON MULTIPLE. Did you ENJOY today's lesson?

Information: Looking through the Lens