The greatest common factor, or GCF, is the largest integer that can divide two or more integers with no remainder. For instance, finding the GCF of 24 and 36 means identifying the greatest whole number both 24 and 36 can be divided by with no remainder. But to properly understand why the greatest common factor of 24 and 36 is 12, it’s important to first develop an understanding of factors and multiples.
What Are Factors and Multiples?
Factors are numbers that can divide into a given number exactly. For instance, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24 because each of these numbers will divide into 24 exactly with no remainder. Meanwhile, multiples are numbers that result from multiplying a given number by another integer. For instance, the multiples of 24 are 24 (1×24), 48 (2×24), 72 (3×24), 96 (4×24), 120 (5×24) etc.
It is important to understand the difference between factors and multiples, as they are often confused. Factors are the numbers that divide into a given number, while multiples are the numbers that result from multiplying a given number by another integer. Knowing the difference between factors and multiples can help you solve math problems more quickly and accurately.
Factors of 24
To work out the GCF of 24 and 36, you’ll need to know the factors of 24. As mentioned above, they are 1, 2, 3, 4, 6, 8, 12, and 24.
The greatest common factor (GCF) of two numbers is the largest number that divides both numbers evenly. In this case, the GCF of 24 and 36 is 12, since 12 is the largest number that divides both 24 and 36 evenly.
Factors of 36
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. To identify the greatest common factor of 24 and 36 you’ll need to uncover which of these numbers can divide both 24 and 36 without leaving a remainder.
The greatest common factor of 24 and 36 is 12. This is because 12 is the largest number that can divide both 24 and 36 without leaving a remainder. To find the greatest common factor, you can also use the prime factorization method. This involves breaking down each number into its prime factors and then finding the common factors between them.
Steps to Calculate the Greatest Common Factor
To calculate the greatest common factor of 24 and 36, you have five steps to complete:
- List the factors of each number
- Compare the lists to identify common factors
- Choose the highest common factor
- Check your answer by dividing one number by the GCF
- Check your answer by dividing the other number by the GCF.
By following these steps completely and accurately, it can be determined that the greatest common factor of 24 and 36 is 12.
It is important to note that the greatest common factor is the largest number that can divide both numbers evenly. This means that the greatest common factor of 24 and 36 is the largest number that can be divided into both 24 and 36 without leaving a remainder.
Understanding Prime Factorization
Another method for determining the greatest common factor of two or more numbers is to use prime factorization. Prime factorization is the act of breaking down a number into its prime factors. To start, two or more numbers must be identified and listed in descending order of size from largest to smallest. Once done, their prime factors must be listed. To do this, you’ll need to identify all possible factors for each a number and then continue to divide these factors into prime numbers until you’re left with a unique combination of prime numbers.
Using the Prime Factorization Method to Find the Greatest Common Factor
Once you have a list of each number’s prime factors, you’re ready to work out the GCF. To do this locate any common prime factors and then multiply them together. The result will be the greatest common factor.
Other Ways to Find the Greatest Common Factor of 24 and 36
Aside from prime factorization and using the five steps outlined above, there are several other ways to find the GCF of two or more numbers. These include using a prime factor tree diagram or Euclidean Algorithm. Meanwhile, a calculator may be used to uncover the GCF quickly and accurately.
Applications for Finding the Greatest Common Factor
In mathematics finding the GCF is important for a range of applications. For instance it can be used to find fractional equivalents or reduce fractions. It’s also helpful for problem solving in algebra. But beyond mathematics, the greatest common factor can be applicable in everyday situations. For instance if you have items (like plates) that can only be shared in whole numbers, you’ll need to work out the GCF to evenly divide them among people.
