How to Find Factors of Numbers?

A factor is a Latin word, meaning “a doer” or “a maker” or “a performer.” We can use the multiplication as well as the division method, to find the factors.

Related Topics

• How to Find the Greatest Common Factor (GCF)
• How to Factor Trinomials

Step by step guide to factoring numbers

In mathematics, a factor is a number that divides another number evenly, that is without remainder. The factors of a number can be called numbers or algebraic expressions that evenly divide a given number/expression. The factors of a number can either be positive or negative.

Properties of factors

Factors of a number have several properties. The following are the properties of the factors:

• The number of factors of a number is finite.
• A factor of a number is always less than or equal to the given number.
• Every number except \(0\) and \(1\) has at least two factors, \(1\) and itself.
• Division and multiplication are operations used to find factors.

How to find factors of a number?

We can use “division” and “multiplication” to find the factors.

• Factors by division

To find the factors of a number using division:

1. Find all the numbers less than or equal to the given number.
2. Divide the given number by each of the numbers.
3. The divisors that give the remainder to be \(0\) are the factors of the number.

For example: Find the positive factors of \(6\) using division.

Positive numbers that are less than or equal to \(6\) are \(1, 2, 3, 4, 5,\) and \(6\). Let \(6\) be divided by each of these numbers.

We can see that the divisors of \(1, 2, 3,\) and \(6\) give zeros as a remainder. Therefore, factors \(6\) are \(1, 2, 3,\) and \(6\).

• Factors by multiplication

To find the factors using the multiplication:

1. Write the given number as the product of two numbers in different possible ways.
2. All the numbers that are involved in all these products are the factors of the given number.

For example: Find the positive factors of \(24\) using multiplication.

We will write \(24\) as the product of two numbers in multiple ways.

All the numbers that are involved in these products are the factors of the given number (by the definition of a factor of a number). So, the factors of \(24\) are \(1, 2, 3, 4, 6, 8, 12,\) and \(24\).

How to find the number of factors?

Using the following steps we can find the number of factors of a given number.

• Step 1: Find its prime factorization, that is, express it as the product of primes.
• Step 2: Write the prime factorization in the exponent form.
• Step 3: Add \(1\) to each of the exponents.
• Step 4: Multiply all the resultant numbers. This product would give the number of factors of the given number.

Factoring Numbers – Example 1:

Find the number of factors of the number \(108\).

Solution:

Do prime factorization of the number \(108\):

Therefore, \(108=2×2×3×3×3\). In the form of power: \(108= 2^2× 3^3\). Add \(1\) to each of the exponents, \(2\) and \(3\). Then, \(2+1=3, 3+1=4\). Multiply these numbers: \(3×4=12\). Therefore, the number of factors \(108\) is \(12\).

The actual factors of \(108\) are \(1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54,\) and \(108\). Here, \(108\) has \(12\) factors and hence our above answer is correct.

Exercises for Factoring Numbers

Find the positive factors of each number.

1. \(\color{blue}{80}\)
2. \(\color{blue}{132}\)
3. \(\color{blue}{270}\)
4. \(\color{blue}{450}\)

1. \(\color{blue}{1,2,4,5,8,10,16,20,40,80}\)
2. \(\color{blue}{1,2,3,4,6,11,12,22,33,44,66,132}\)
3. \(\color{blue}{1,2,3,5,6,9,10,15,18,27,30,45,54,90,135,270}\)
4. \(\color{blue}{1,2,3,5,6,9,10,15,18,25,30,45,50,75,90,150,225,450}\)