What do Factors of a Number imply?
In mathematics, the factor of a number refers to the complete division of a number with no trace of remainders.
Common Factors
It is possible for two or more numbers to have common factors. For example, if we consider the numbers 16 and 20, then,
The factors of 16 are 1, 2, 4, 8, and 16.
The factors of 20 are 1, 2, 4, 5, 10, and 20.
The common factors can be encircled as shown in the figure below:
Thus, the common factors of 16 and 20 are 1, 2, and 4.
Prime Factors
Prime factors refer to the factors of prime numbers. For example, 2, 3 and 5 are prime factors of 20.
There are innumerable ways to find prime factors of a number. However, the most common is the factor tree.
The factor tree of 20 can be drawn as follows:
Greatest Common Factor
The greatest among the common factors of two numbers is known as the Greatest Common Factor. Its abbreviated form is GCF. The greatest common factor can be found for two or more numbers. The most frequently-used methods to find the GCF are the listing method and the prime factorisation method.
Let’s look at some examples to understand this better.
Examples of the listing Factor Method
Example 1: Find the GCF of 36 and 42.
Solution:
The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, and 42.
The common factors have been encircled below:
The common factors of 36 and 42 are 1, 2, 3, and 6. Hence the GCF of 36 and 42 is 6.
Example 2: Find the GCF of 14 and 21.
Solution:
The factors of 14 are 1, 2, 7, and 14.
The factors of 21 are 1, 2, 3, 7, and 21.
The common factors have been encircled below:
The common factors of 14 and 21 are 1, 2, and 7. Hence, the GCF of 14 and 21 is 7.
An example of the prime factor method:
Example 3: Find the GCF of 9 and 18.
Solution:
The factor tree of 9 can be drawn as follows:
The factor tree of 18 is as follows:
The prime factorization of 9 and 18 is as follows:
9 = 3 \( \times \) 3
18 = 2 \( \times \) 3 \( \times \) 3
Now highlight the common factors:
9 = 3\(\times\)3
18 = 2 \( \times \) 3\( \times \) 3
The GCF is the product of the common prime factors, that is,
3 \( \times \) 3 = 9.
An example of how to find two numbers with a given GCF:
Example 4: Out of the given pair of numbers, which one has the GCF of 8?
A) 64, 16 B) 32, 14 C) 6, 38 D) 96, 18
Solution :
The number 14 has 7 and 2 as its factors. Since 8 is not a factor of 14, option B is incorrect.
The number 8 cannot be a factor of 6 because 6 is lesser in value than 8. Hence, option C is also incorrect.
The factors of 18 are 1, 2, 3, 6, 9, and 18. Since it does not include 8, option D is incorrect as well.
Option A:
The factors of 64 are 1, 2, 4, 8, 16, 32, and 64
The factors of 16 are 1, 2, 4, 8, and 16
The common factors and the GCF have been encircled below:
A real-life Modeling Example
21 cantaloupes and 49 watermelons are equally packed in cartons so that no fruit is left. What is the maximum or the highest possible number of cartons needed?
Solution:
To find the maximum or the highest possible number of cartons needed, we need to find the GCF of 21 and 49.
The prime factorisation of 21 and 49 is given below and the common prime factors have been encircled :
As shown above, the GCF is 7.
Hence, 7 cartons are needed.
GCF, or the greatest common factor, is the greatest of all common factors between any two numbers.
GCF is the greatest common factor, whereas LCM is the least common multiple of two or more numbers.