This video will show you strategies for the GMAT Greatest Common Divisor. Watch, Learn, and Ace the GMAT Exam.
GMAT Greatest Common Divisor Strategies
In this lesson, we’ll learn all about greatest common divisors. To begin, the greatest common divisor is the same as the greatest common factor, so you’ll see these two terms used interchangeably. Now the greatest common divisor is the greatest positive divisor shared by two or more numbers. To illustrate this concept, let’s find the greatest common divisor of 12 and 30. Well, here are the positive divisors of 12, and here are the positive divisors of 30. Notice that 12 and 30 have several divisors in common. They both share one as a common divisor, and they share a common divisor of two, three, and six. Now of all of the divisors they have in common, the greatest is six. So the greatest common divisor of 12 and 30 is 6.
Let’s try one more example. To find the greatest common factor of 56 and 70, we’ll list the divisors of 56 and the divisors of 70. As you can see, the divisors that they have in common are 1, 2, 7, and 14. So, the greatest common factor of 56, and 70 is 14. As you’ve just seen, one method for finding the greatest common divisor is to list all of the positive divisors, and then determine which of the common divisors is the greatest. Now this method can be somewhat cumbersome when dealing with large numbers.
So let’s look at another technique for finding the greatest common divisor. To demonstrate this technique, we’ll find the greatest common divisor of 56 and 70. Now to begin, we’ll find the prime factorization of 56, and the prime factorization of 70. At this point, we’ll find all of the prime factors that they have in common. Now they both share a two, so we’ll note that. And they also share a seven, so we’ll note that. So 56, and 70 share a 2, and a 7 in their prime factorization. So the greatest common divisor of 56 and 70, will be the product of 2 and 7 which is equal to 14.
Let’s do one more. Let’s find the greatest common divisor of 132, 198, and 330. Now keep in mind that it is very rare to have to find the greatest common divisor of more than two numbers. However, the technique for three numbers is the same as the technique for two numbers. So let’s proceed. First, we’ll find the prime factorization of each number. Then we’ll identify all of the prime factors that all three numbers have in common, and finally the greatest common divisor will be the product of the shared prime factors.
So the greatest common divisors of the three numbers here is 66. So that’s the systematic way to fine the greatest common divisor. This technique works for large numbers, and it is also very useful for getting insight into questions involving greatest common divisors. Now you might be wondering, what happens if you have two numbers that do not share any prime factors. For example, let’s say you want to find the greatest common divisor of 8 and 15. Using the prime factorization technique, we’ll first find the prime factorization of 8, and the prime factorization of 15.
Notice that 8 and 15 do not share any prime factors. So what then is the greatest common divisor here? When two or more numbers do not have any prime factors in common, the greatest common divisor will equal one. The reason for this, is that although it isn’t necessarily apparent here, all positive integers have one as a divisor. So if we list the positive divisors of 8 and 15, we can see that both numbers have 1 as a common divisor. So the greatest common divisor of 8 and 15 is 1.
Now, if you want to become an expert on greatest common divisors, you must develop certain skills. First, you must be able to find greatest common divisors by listing divisors as we did in the very first example. You should also be good at finding greatest common divisors in your head. For example, what is the greatest common divisor of 10 and 25? In other words, what is the largest divisor that 10 and 25 both have in common? It’s five. What about the greatest common divisor of 18 and 45? Did you get nine? How about 20 and 64? The greatest common divisor here is four. Another important skill you need, is the ability to find the greatest common divisor using the prime factorization technique.
And finally, you must be good at selecting pairs of numbers that have a certain greatest common divisor. For example, what are two numbers with the greatest common divisor of two? Now there are a lot of number pairs that meet this condition, can you think of any? Well, how about two and four? They have a greatest common divisor of two. Or perhaps we could have chosen 6 and 22. Now, what are the two smallest positive numbers that have a greatest common divisor of two? Did you get two and two? Their greatest common divisor is two.
Let’s try one more. What are two numbers that have a greatest common divisor of 15? Well, how about 15 and 15. Or 15 and 30? Or perhaps even 45 and 120? There are many more pairs of numbers that have a greatest common divisor of 15, so see if you can find some more.
If you can master all of these skills, you will be well on your way to becoming an expert at greatest common divisors.