Common Factors

How do I find a common factor?

To understand this term, we first have to define factors. A simple definition of factors is that they are whole numbers whose product gives another number. In other words, if a is a factor of A, then dividing A by should leave no remainder.

Prime factorization is the method that gives factors of a number. There are many ways of doing it, but we will only focus on two of them.

Method 1: Table form

It is easy to tabulate your data if you are finding factors of a given number. For example, what are the factors of 128?

 1 128 128 ​ ​ ​ ​

The yellow column is where factors of 128 will appear. By beginning with 1 as the first factor of any number, you grow the table by dividing the 128 by one and entering the result in the row below 128. The next step is to divide the resulting number by 2 to get 64. If the number is not divisible by 2, you go to the next digit which is 3. Continue with the same procedure until you have only 1 on the right column. Here is a completed table.

 1 128 2 128 2 64 2 32 2 16 2 8 2 4 2 2 ​ 1

The factors of 128 are all in the yellow column. 128 = 2×2×2×2×2

Example 2: Find the factors of 126

Solution

 1 126 2 126 3 63 3 21 7 7 1

Factors of 126 are the numbers in the yellow column. 126 = 2×3×3×7

Method 2: Flow chart method

The principle is the same, only that we use a flow chart instead of tables. Let’s illustrate this using example.

Find the factors of 126.

Solution

Values in the yellow ovals are the factors of 126. This method applies to any number that you will come across. It should be fast sketching the ovals.

How Do I Find Common Factors?

To find common factors for a given set of numbers, you first have to find the individual factors. For example, find common factors of 126 and 56.

Using any of the methods above, the factors of the given numbers are:

126 = 2×3×3×7

56 = 2×2×2×7

Common factors for 126 and 56 are 2 and 7. We get these values by looking at factors that are common to both 56 and 128.

What is Greatest common factor?

The greatest common factor (aka Highest common factor) is simply the greatest factor that we can find from both numbers. For example, the greatest common factor of 126 and 56 is 7. We follow the same procedure of finding factors and identifying the common factors.

Example:

Find the greatest common factors of 50 and 90.

Solution

We start by listing out the factors. Using any of the methods that we already know or any other that you are familiar with, the factors are:

50 = 2×5×5

90 = 2×3×3×5

Common factors for 50 and 90 are 2 and 5. We can easily see that 5 is greater than 2, and hence, the greatest common factor.

Remarks

The knowledge of finding the greatest common factor of a number helps simplify fractions. To see that in action, get to our other tutorial about fraction arithmetic or simplification of algebraic fractions. You will use it many times in later topics. Therefore, having a good foundation in it is particularly important. Always remember to find the prime factors first, after which you can get common factors and the greatest common factor.

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