GCF and LCM: The Greatest Common Factor and Least Common Multiple

gcf and lcmThese are important math concepts as is, but if you’re looking to take the GRE or GMAT, you’ll need to know how to  find the greatest common factor and least common multiple. This is otherwise known as “prime factorization.”

As the name it implies, prime factorization is finding the prime factors of a number. What are prime numbers? Prime numbers are numbers that are only divisible by 1 and itself. The first prime number is 2. 2 is only divisible by integers 1 and 2. The next prime number would be 3. 3 is only divisible by 1 and 3. 4 would not be a prime number, because it’s divisible by 2. 5 would be a prime number. 6 would not be a prime number because it’s also divisible by 2 and 3 and if I keep that pattern up, I can come up with quite a few prime numbers.

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Let’s talk about factors

Factors of a number are the numbers that multiply together to give the number itself. For example, the factors for 6 can be 2 and 3. The factors for 6 can also be 1 and 6. All of the factors for 6 are 1, 2, 3, and 6. When you do the prime factorization of a number, you find all of the factors of a number and you keep splitting them apart until you only have prime numbers. For example, let’s do the prime factorization for 18. First, you know that 18 is 3 x 6. Six is not a prime number. I can split that apart a little bit more so I still have the first 3 and the 6’s 3 and 2. So, the prime factorization of 18 is 3 x 3 x 2.

Now, let’s take 24. 24 is 3 x 8. 8 is not a prime number. It can be split apart into 4 and 2. 4 is not a prime number and that 4 can be split apart into two more 2s and you keep this last 2 as part of the prime factor list. So, the prime factorization of 18 is 3 x 3 x 2. The prime factorization of 24 is 3 x 2 x 2 x 2.

Prime factorization

Now, prime factorization has many practical uses. One good example is finding the greatest common factor for two numbers. Let’s find the greatest common factor between 18 and 24. First, find the prime factorization of 18 and 24. You know that it’s 3 x 3 x 2 for 18 and 24 is 3 x 2 x 2 x 2. So, what is the greatest common factor for 18 and 24? The greatest common factor is a number which has factors that are in both 18 and 24. As the word ‘common’ implies, the factors need to be in both numbers. First, look for any numbers that are in both the prime factorization of 18 and the prime factorization of 24. You can look at each number one-by-one.

First, you’ll see that they both have a 3. Cross that off so you don’t count it again and have one 3 listed. Next, check to see if the 24 has a second 3. It doesn’t, so you can’t use the second 3 as one of the greatest common factors. The 18 has a 2 though and the 24 has a 2. You’ve checked all the numbers, and the two listed for both 18 and 24 are 3 and 2. This means that 3 and 2 are common factors, so 3 x 2 equals 6. The greatest common factor of 24 and 18 is 6.

This process is also useful for simplifying fractions. For example, if you have the fraction 18 over 24 (18/24). You could do the prime factorization of 18 and 24 to find the numbers’ greatest common factor. You then express them both as products of each number’s greatest common factor. You see that 18 has 6, 3 and a 2. 18 has a 6 and the 3 the remains. 24 has a 6 and 2 x 2 or 4. For simplifying fractions, you can find the greatest common factor and divide each number. Since the greatest common factor for 18 and 24 is 6, then 18/24 simplified is 3/4.

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Least common multiple

You can also find the least common multiple of two numbers by using the prime factorization. So, let’s find the least common multiple for 18 and 24. The prime factorization of 18 is 3 x 3 x 2 and the prime factorization of 24 is 3 x 2 x 2 x 2. Now, you can find the least common multiple between 18 and 24.

The best way to explain the least common multiple is by example. First, you choose one of the numbers arbitrarily. For instance, choose the 18 (you could choose the 24 as well). Write the prime factorization for 18 into the least common multiple. So, 18 is 3 x 3 x 2. Now, ask yourself, “Can I find a 24 in here?” Go through one number at a time. First, ask yourself, “Can I find a single 3?” Yes, you can find a single 3. Next, see if you can find a single 2. Yes, there is a single 2. And so the least common multiple of 18 and 24 is 3 x 3 x 2 x 2 x 2 or 18 x 4 or you could view it as 24 x 3. If you multiplied these numbers together, you’d get 72 as your answer. This is the least common multiple.

You might wonder what least common multiples are good for.  You can use them when adding or subtracting fractions from each other, because you need to have a common denominator between them. For example, let’s say that you have 1/18 – 1/24.  Before you can subtract these two numbers without using a calculator, you first need to find the least common multiple of 18 and 24 and change the denominators to that least common multiple. You’ve already calculated the least common multiple as is 72, which is 18 x 4. You can then multiply the 1/18 by 4/4 and I see that the least common multiple of 72 is the 24 x 3. So, you can multiply 1/24 by 3/3. Now, after you’ve made the calculations, you have 4/72 – 3/72. Notice you now have the same number in the denominator, so now you can properly subtract your fraction without using a calculator. The denominator number stays the same, but you subtract the numerator, which is the top number in the fraction. You subtract 3 from 4 and get one, so the answer to the problem is 1/72.

To quickly recap, prime numbers are numbers that are only divisible by 1 and itself. 2 is defined as the first prime number. You can find the greatest common factor by using the prime factorization of two numbers and the greatest common factor consist of the factors that are common, hence, the word greatest common factors that are common to both numbers. With the prime factorization for 18 and 24, I sought that one 3 and one 2 and these values are useful simplifying fractions.

You can also use prime factorization for finding the least common multiple between two numbers. Here, we used the numbers 18 and 24. You multiply the factorization numbers to get 4 x 18 or 3 x 24, which are both equal to 72. The least common multiple has a common application of adding or subtracting fractions from each other. The least common multiple for 18 and 24 is 72. 18 x 4 gives me 72, so you multiply the 1/18 by 4/4. 24 x 3 is 72 and multiply the 1/24 by 3/3 and with a little bit more arithmetic, you get 4/72-3/72. As soon as you have the same common denominator, you subtract the 3 from the 4 to get 1. And thus, ends the lecture on prime factorization, greatest common factor and least common multiple.

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