## Greatest Common Factor (GCF)

#### Result Summary:

 Type Result Given set of numbers 0 Greatest Common Factor (GCF) 0

### What is GCF?

A factor is a number which when multiplied with other numbers gives a larger number. Any natural number can be broken down into its factors: these factors indicate all the possible ways we can multiply them to get the number itself. For example, the factors of 21 are 1, 3, 7 and 21. Every number has 1 and the number itself as its factors. Prime numbers have only 2 factors: 1 and the number itself. Composite numbers are those which have factors other than these two.<.p>

In mathematics, for a given set of two or more numbers, the greatest common factor (GCF), also known as the greatest common divisor (GCD) or highest common factor (HCF) is the largest possible factor that divides all the numbers in this set to result in a natural number. In other words, the GCF is the largest factor that divides all these numbers without leaving a remainder.

Consider the numbers 8, 12, 20. The factors of 8 = 2 × 2 × 2, factors of 12 = 2 × 2 × 3, factors of 20 = 2 × 2 × 5. There are two 2’s common to each of these numbers. So, the GCF is the product of these 2 numbers, 2 × 2 = 4. Hence, the GCF of 8, 12, 20 is 4.

For example, consider the numbers 15 and 21. The factors of 15 are 1, 3, 5, 15 and those of 21 are 1, 3, 7, 21. For finding the GCF, we consider only the prime factors and neglect the 1 and the number itself. So, the factors of 15 = 3 × 5 and 21 = 3 × 7. Here, 3 is common in the list of factors for both these numbers. It is also the largest number which divides both these numbers. Hence, 3 is the GCF of 15 and 21.

### Properties of GCF

Given below are some important properties of the GCF.

Factor – The factor of any number is a smaller number that divides it without any remainders. Every natural number has 1 and the number itself as its factors.

Prime factor – The prime numbers which evenly divide a number. For finding the GCF, it helps to first list out all the prime factors and then look for the ones common to all the numbers under consideration.

GCF – The largest positive integer which is a factor for each of the given numbers and evenly divides them.

### Finding the GCF

There are two ways to find the GCF of a given set of numbers.

1) Factoring:. To find the GCF of some numbers, list out all the factors of these numbers. The integer factors are those which divide each of these numbers without any remainder. When the factors of the numbers are listed, the GCF is the largest number common to each list.

For example, find the GCF of 20, 60, 150.

20 = 1, 2, 4, 5, 10, 20.

60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

150 = 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150.

Hence, the GCF is 10.

2) Prime factoring: To find the GCF, prime factoring is another method. In this case, list out only the prime factors of each number (even if they are repeated). Make a note of the prime factors which are common to each number. Include the highest number of times a prime factor occurs, that is common to each of the numbers. Multiply these to obtain the GCF.

In some cases, the prime factoring might be easier than the factoring method.

For example, find the GCF of 20, 60, 150.

20 = 2 × 2 × 5.

50 = 2 × 5 × 5.

150 = 2 × 3 × 5 × 5.

The common factors in each number are 2 and 5. Hence, the GCF = 2 × 5 = 10.

### Characteristics of GCF

• The GCF of any number and 0 is the number itself. For example, GCF (9, 0) = 9. But GCF (0, 0) = undefined.
• The GCF of two or more numbers is always smaller than those numbers.
• The GCF of some numbers, where one of the numbers is a prime, is either 1 or the prime number. For example, the GCF (3, 9) = 3, the GCF (2, 11) = 1.
• The GCF of any two consecutive numbers is 1. For example, GCF (13, 14) = 1, GCF (8, 9) = 1.
• If the numbers are all primes, then the GCF is 1. For example, GCF (11, 13) = 1, GCF (3, 5, 7) = 1.

• ### Areas of application

The GCF is frequently used in the following situations:

• To split something into smaller parts.
• To equally distribute a number of items into their largest groups.
• To arrange something in rows or columns or groups.
• Given below is a real-life use of GCF.

Question: A middle school teacher needs to arrange some students in the least number of rows for a marchpast. There are 21 students from grade 5, 56 students from grade 6 and 35 students from grade 7. How can she arrange the students so that the students in each row belong to the same grade?

Answer: The number of students that need to be arranged from grades 5, 6 and 7 are 21, 56 and 35 respectively. To do this, we can find the GCF of the numbers 21, 56 and 35.

21 = 3 × 7, 56 = 2 × 2 × 2 × 7, 35 = 5 × 7.

The GCF of these numbers is 7.

This means that 3 rows of grade 5, 8 rows of grade 6 and 5 rows of grade 7 students can be arranged, and the number of students in each row would be 7.

The total number of rows is 3 + 8 + 7 = 18.