by Josh Rappaport
In last month’s article, I presented a quick-and-easy way to find the Least Common Multiple (LCM, aka LCD) for any pair of numbers.
In this month’s article, I’d like to provide a similarly quick-and-easy way to find the LCM’s close cousin, the Greatest Common Factor (GCF, aka GCD).
As you may distantly remember (elementary school and middle school seem so far away, do they not?), the GCF refers to the largest number that divides evenly into two or more numbers. So, for example, given the numbers 6 and 8, the GCF is 2 since 2 is the largest number that divides evenly into 6 and 8. And given the numbers 12 and 30, the GCF is 6, since 6 is the largest number that divides evenly into 12 and 30.
So what I’m going to do here is show you how to find the GCF for any two numbers quickly and easily.
To start, I need to introduce a new concept, one I made up by myself, I’m somewhat proud to say. And this new concept is what I call the GPGCF. The what, you ask? The GPGCF, which stands for the “Greatest Possible Greatest Common Factor.” O.K., now that you are thoroughly annoyed at me for giving you a “five-letter abbreviation, let me assure that it really is helpful. And, if you find it annoying to reel off five letters of the alphabet, I won’t get offended if you shorten the abbreviation to just: GPG. In fact I’ll use GPG and GPGCF interchangeably in this piece, just to show you what a flexible kind of guy I am.
Back to the main thread, though, what in the world is the GPGCF? Well, the GPGCF is an upper limit for the GCF. And that’s a really nice thing to know because if you don’t have an upper limit, you probably feel unsure as to when you can stop looking for the GCF. For example let’s say you’re seeking the GCF for 48 and 60. You find out that 12 divides evenly into both numbers, and you have a hunch that 12 is the GCF. But who’s to say that 12 is for sure the highest possible GCF? So you keep seeking, looking through the teen numbers and through the 20s and 30s. And then, after you’ve checked all of the numbers possible, you discover that your hunch was right; 12 is the GCF. But how annoying! You wasted all of that time looking, all because you didn’t know when you could stop looking.
But starting today, you will know when you CAN stop looking, since the GPG tells you where that number is, since the GPG IS the upper limit for this search.
Not only that, another nice thing about the GPG is that it also clues you in as to which specific numbers are authentic candidates for being the GCF. It turns out, for example, that with the numbers 48 and 60, there are only five numbers that are actual candidates for being the GCF. And we learn this through the GPG.
Finally, using the GPG, you will discover those candidate numbers in order from largest to smallest, and you’ll test them in order from largest to smallest. That way, once you find the first candidate that does divide evenly into both numbers, you’ll know that you have found the true, the authentic, the “real McCoy” GCF. Again, all of these things save you time and trouble, and they give you considerably more power over numbers.
So with such a laudatory intro, you might be wondering how we actually find the GPG and how, using the GPG, we find the real candidates.
Turns out that it’s pretty simple. You simply look at the two numbers for which you’re seeking the GCF, then ASK and ANSWER THESE THREE QUESTIONS:
Q #1) Of the two numbers, which is the smaller?
Q #2) Given the two numbers, what is their difference?
Q #3) What’s smaller, your answer to Q #1 or your answer to Q #2? Whichever is smaller, that’s the GPG.
So this is all fine and good, but don’t you think you’d understand this better if we showed this idea through a real example? Let’s do so, seeking the GCF for 48 and 84. Here’s how we answer the questions:
Q #1) Which is smaller, 48 or 84?
A: 48
Q #2) What’s the difference between 48 and 84.
A: 36, since 84 – 48 = 36. Simple subtraction … that’s all.
Q#3) What’s smaller: the answer to Q #1 (48), or the answer to Q #2 (36).
A: 36 is smaller … obviously.
That means that the GPGCF is 36!
Next question, given that 36 is the upper limit of the GCF for 48 and 84, how do you find the “actual candidates for the GCF”?
It turns out that you find the candidates by finding the FACTORS of the GCF. So in this example, we find the factors of 36 and write them in decreasing order.
The factors of 36, then, are: 36, 18, 12, 9, 4, 3 and 2 … just seven factors. Then we start testing the numbers to see which of them divides evenly into both 48 and 84. Here’s the test …
36 does NOT divide evenly into 48, so it is out.
18 does NOT divide evenly into 48 either, so it is out.
But 12 DOES divide evenly into both 48 and 84, and here’s how we know.
48 ÷ 12 = 4, and 84 ÷ 12 = 7, with 4 and 7 both being whole numbers.
Since 12 is the largest candidate number that divides evenly into both numbers, 12 is the GCF. No more searching. You are DONE!
And now, to help everyone get this process down solidly, let’s do it for one more example. Suppose you want to find the GCF for 18 and 90. Here are the steps, with the actual work shown.
Q #1) Which is smaller, 18 or 90?
A: 18
Q #2) What’s the difference between 18 and 90?
A: 72, since 90 – 18 = 72. Simple subtraction.
Q#3) What’s smaller: the answer to Q #1 (18), or the answer to Q #2 (72).
A: 18 is smaller … obviously.
So this means that 18 is the GPGCF.
To find the GCF, then, first we list the factors of 18, in decreasing order. They are: 18, 9, 6, 3 and 2.
Then we start testing these numbers. First 18. Well 18 obviously does divide into 18, and 18 also does divide into 90. So that means that 18 must be the GCF! This time we found the GCF on our first try.
And one of the nice things about this technique is that you can find the GCF on your first try. Not always, of course, but it does happen.
To help everyone master this approach, I’ve set up four problems for you to do. Here are the directions, problems, and answers.
1) Find GPGCF and say if it is the difference or the smaller number.
2) List the factors of the GPGCF, greatest to least.
3) Find the GCF by testing the factors from greatest to least.
PROBLEMS:
a) 8 and 12
b) 16 and 40
c) 18 and 63
d) 56 and 140
ANSWERS:
a) 8 and 12: GPGCF = 4 (difference)
Factors of 4: 4 and 2
GCF = 4
b) 16 and 40: GPGCF = 16 (smaller number)
Factors of 16: 16, 8, 4 and 2
GCF = 8
c) 18 and 66 : GPGCF = 18 (smaller number)
Factors of 18: 18, 9, 6, 3 and 2
GCF = 6
d) 56 and 76: GPGCF = 20 (difference)
Factors of 20: 20, 10, 5, 4 and 2
GCF = 4
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
In last month’s article, I presented a quick-and-easy way to find the Least Common Multiple (LCM, aka LCD) for any pair of numbers.
In this month’s article, I’d like to provide a similarly quick-and-easy way to find the LCM’s close cousin, the Greatest Common Factor (GCF, aka GCD).
As you may distantly remember (elementary school and middle school seem so far away, do they not?), the GCF refers to the largest number that divides evenly into two or more numbers. So, for example, given the numbers 6 and 8, the GCF is 2 since 2 is the largest number that divides evenly into 6 and 8. And given the numbers 12 and 30, the GCF is 6, since 6 is the largest number that divides evenly into 12 and 30.
So what I’m going to do here is show you how to find the GCF for any two numbers quickly and easily.
To start, I need to introduce a new concept, one I made up by myself, I’m somewhat proud to say. And this new concept is what I call the GPGCF. The what, you ask? The GPGCF, which stands for the “Greatest Possible Greatest Common Factor.” O.K., now that you are thoroughly annoyed at me for giving you a “five-letter abbreviation, let me assure that it really is helpful. And, if you find it annoying to reel off five letters of the alphabet, I won’t get offended if you shorten the abbreviation to just: GPG. In fact I’ll use GPG and GPGCF interchangeably in this piece, just to show you what a flexible kind of guy I am.
Back to the main thread, though, what in the world is the GPGCF? Well, the GPGCF is an upper limit for the GCF. And that’s a really nice thing to know because if you don’t have an upper limit, you probably feel unsure as to when you can stop looking for the GCF. For example let’s say you’re seeking the GCF for 48 and 60. You find out that 12 divides evenly into both numbers, and you have a hunch that 12 is the GCF. But who’s to say that 12 is for sure the highest possible GCF? So you keep seeking, looking through the teen numbers and through the 20s and 30s. And then, after you’ve checked all of the numbers possible, you discover that your hunch was right; 12 is the GCF. But how annoying! You wasted all of that time looking, all because you didn’t know when you could stop looking.
But starting today, you will know when you CAN stop looking, since the GPG tells you where that number is, since the GPG IS the upper limit for this search.
Not only that, another nice thing about the GPG is that it also clues you in as to which specific numbers are authentic candidates for being the GCF. It turns out, for example, that with the numbers 48 and 60, there are only five numbers that are actual candidates for being the GCF. And we learn this through the GPG.
Finally, using the GPG, you will discover those candidate numbers in order from largest to smallest, and you’ll test them in order from largest to smallest. That way, once you find the first candidate that does divide evenly into both numbers, you’ll know that you have found the true, the authentic, the “real McCoy” GCF. Again, all of these things save you time and trouble, and they give you considerably more power over numbers.
So with such a laudatory intro, you might be wondering how we actually find the GPG and how, using the GPG, we find the real candidates.
Turns out that it’s pretty simple. You simply look at the two numbers for which you’re seeking the GCF, then ASK and ANSWER THESE THREE QUESTIONS:
Q #1) Of the two numbers, which is the smaller?
Q #2) Given the two numbers, what is their difference?
Q #3) What’s smaller, your answer to Q #1 or your answer to Q #2? Whichever is smaller, that’s the GPG.
So this is all fine and good, but don’t you think you’d understand this better if we showed this idea through a real example? Let’s do so, seeking the GCF for 48 and 84. Here’s how we answer the questions:
Q #1) Which is smaller, 48 or 84?
A: 48
Q #2) What’s the difference between 48 and 84.
A: 36, since 84 – 48 = 36. Simple subtraction … that’s all.
Q#3) What’s smaller: the answer to Q #1 (48), or the answer to Q #2 (36).
A: 36 is smaller … obviously.
That means that the GPGCF is 36!
Next question, given that 36 is the upper limit of the GCF for 48 and 84, how do you find the “actual candidates for the GCF”?
It turns out that you find the candidates by finding the FACTORS of the GCF. So in this example, we find the factors of 36 and write them in decreasing order.
The factors of 36, then, are: 36, 18, 12, 9, 4, 3 and 2 … just seven factors. Then we start testing the numbers to see which of them divides evenly into both 48 and 84. Here’s the test …
36 does NOT divide evenly into 48, so it is out.
18 does NOT divide evenly into 48 either, so it is out.
But 12 DOES divide evenly into both 48 and 84, and here’s how we know.
48 ÷ 12 = 4, and 84 ÷ 12 = 7, with 4 and 7 both being whole numbers.
Since 12 is the largest candidate number that divides evenly into both numbers, 12 is the GCF. No more searching. You are DONE!
And now, to help everyone get this process down solidly, let’s do it for one more example. Suppose you want to find the GCF for 18 and 90. Here are the steps, with the actual work shown.
Q #1) Which is smaller, 18 or 90?
A: 18
Q #2) What’s the difference between 18 and 90?
A: 72, since 90 – 18 = 72. Simple subtraction.
Q#3) What’s smaller: the answer to Q #1 (18), or the answer to Q #2 (72).
A: 18 is smaller … obviously.
So this means that 18 is the GPGCF.
To find the GCF, then, first we list the factors of 18, in decreasing order. They are: 18, 9, 6, 3 and 2.
Then we start testing these numbers. First 18. Well 18 obviously does divide into 18, and 18 also does divide into 90. So that means that 18 must be the GCF! This time we found the GCF on our first try.
And one of the nice things about this technique is that you can find the GCF on your first try. Not always, of course, but it does happen.
To help everyone master this approach, I’ve set up four problems for you to do. Here are the directions, problems, and answers.
1) Find GPGCF and say if it is the difference or the smaller number.
2) List the factors of the GPGCF, greatest to least.
3) Find the GCF by testing the factors from greatest to least.
PROBLEMS:
a) 8 and 12
b) 16 and 40
c) 18 and 63
d) 56 and 140
ANSWERS:
a) 8 and 12: GPGCF = 4 (difference)
Factors of 4: 4 and 2
GCF = 4
b) 16 and 40: GPGCF = 16 (smaller number)
Factors of 16: 16, 8, 4 and 2
GCF = 8
c) 18 and 66 : GPGCF = 18 (smaller number)
Factors of 18: 18, 9, 6, 3 and 2
GCF = 6
d) 56 and 76: GPGCF = 20 (difference)
Factors of 20: 20, 10, 5, 4 and 2
GCF = 4
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
