# The Greatest Common Factor

The greatest common factor, or GCF, of two or more integers (...,-2, -1, 0, 1, 2, ...)  is the largest natural number (1, 2, 3, ...) that will divide evenly into all of the integers a natural number of times.

It is important to note the word common embedded in the term greatest common factor.  This implies, of course, that the two numbers have something in common.  This concept can best be seen by making factor trees.  Make factor trees for the two numbers in number four above.

It would probably not be an efficient use of time to make factor trees for every GCF problem.  Students should use their knowledge of multiplication and division, as well as the calculator, in order to complete the problems in a timely fashion.

–Strategy 1: Find the prime factorization of each number – Take whatever they have in common (to the highest power possible)
•    Find the GCF(42, 385) Factorization 42=2⋅21=2⋅3⋅7
Factorization 385=5⋅77=5⋅7⋅11

GCF(42, 385) = 7^1 = 7

•   Find the GCF(338, 507)Factorization 338 = 2⋅169 = 2⋅13^2

Factorization 507 = 3⋅169 = 3⋅13^2

GCF(338, 507) = 13^2 = 169

Find the GCF (greatest common factor) of the following numbers using a Factor Tree:

Strategy 2: list all of the common factors of 2 numbers and take the largest.
Ex    GCF of 36 and 84
Factors of 36:    1, 2, 3, 4, 6, 9, 12, 18, 36                   Factors of 84:    1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
GCF (36, 84) = 12
The Venn Factor: In this lesson, students use a Venn diagram to sort prime factors of two or more positive integers. Students calculate the greatest common factor by multiplying common prime factors and develop a definition based on their exploration.

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### Figure This Challenge #56

• Complete Solution will be given on May 17, 2015

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Created by Wanda Collins May 10, 2015 at 1:56pm. Last updated by Wanda Collins May 10, 2015.

# Math Limerick

Question: Why is this a mathematical limerick?

( (12 + 144 + 20 + 3 Sqrt[4]) / 7 ) + 5*11 = 92 + 0 .

A dozen, a gross, and a score,
plus three times the square root of four, divided by seven, plus five times eleven, is nine squared and not a bit more.

---Jon Saxton (math textbook author)

Presentation Suggestions:
Challenge students to invent their own math limerick!

The Math Behind the Fact:
It is fun to mix mathematics with poetry.

Resources:

Su, Francis E., et al. "Math Limerick." Math Fun Facts.

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