Factors

A factor is a Latin word, and it means "a doer" or "a maker" or "a performer." A factor of a number in math is a number that divides the given number. Hence, a factor is nothing but a divisor of the given number. To find the factors, we can use the multiplication as well as the division method. We can also apply the divisibility rules.

Factoring is a useful skill to find factors, which is further utilized, in real-life situations, such as dividing something into equal parts or dividing in rows and columns, comparing prices, exchanging money and understanding time, and making calculations, during travel.

In math, a factor is a number that divides another number evenly, that is with no remainder. Factors can be algebraic expressions as well, dividing another expression evenly. Well, factors and multiples are a part of our daily life, from arranging things, such as sweets in a box, handling money, to finding patterns in numbers, solving ratios, and working with expanding or reducing fractions.

Factor Definition

A factor is a number that divides the given number without any remainder. Factors of a number can be referred to as numbers or algebraic expressions that evenly divide a given number/expression. The factors of a number can either be positive or negative.

For example, let's check for the factors of 8. Since 8 can be factorized as 1 × 8 and 2 × 4 and we know that the product of two negative numbers is a positive number only. Therefore, the factors are 8 are actually 1, -1, 2, -2, 4, -4, 8 and -8. But when it comes to problems related to the factors, only positive numbers are considered, that too a whole number and a non-fractional number.

Factors of a number have a certain number of properties. Given below are the properties of 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 the operations that are used in finding the factors.

We can use both "Division" and "Multiplication" to find the factors.

Factors by Division

To find the factors of a number using division:

• Find all the numbers less than or equal to the given number.
• Divide the given number by each of the numbers.
• The divisors that give the remainder to be 0 are the factors of the number.

Example: Find the positive factors of 6 using division.

Solution:

The positive numbers that are less than or equal to 6 are 1, 2, 3, 4, 5, and 6. Let us divide 6 by each of these numbers.

We can observe that divisors 1, 2, 3, and, 6 give zero as the remainder. Thus, factors of 6 are 1, 2, 3, and 6.

Factors by Multiplication

To find the factors using the multiplication:

• Write the given number as the product of two numbers in different possible ways.

• All the numbers that are involved in all these products are the factors of the given number.

Example: Find the positive factors of 24 using multiplication.

Solution:

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)

Thus, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

We can find the number of factors of a given number using the following steps.

• Step 1: Find its prime factorization, i.e. express it as the product of primes.
• Step 3: 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.

Example: Find the number of factors of the number 108.

Solution:

Perform prime factorization of the number 108:

Thus, 108 = 2 × 2 × 3 × 3 × 3. In the exponent form: 108 = 2² × 3³. Add 1 to each of the exponents, 2 and 3, here. Then, 2 + 1 = 3, 3 + 1 = 4. Multiply these numbers: 3 × 4 = 12. Thus, Number of factors of 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.

Factors do exist for an algebraic expression as well. For example, the factors of 6x are 1, 2, 3, 6, x, 2x, 3x, and 6x. There are different types of procedures to find factors in algebra. Some of them are as follows:

• Factoring Expressions
• Factoring Quadratic Expressions
• Factoring Polynomials

We will learn about these types of factoring in higher grades. Click on the above links to learn each of them in detail.

Factors of Numbers

Given below is the list of topics that are closely connected to Factors. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

What are Factors in Math?

A factor is a number that divides the given number without any remainder. The factors of a number can either be positive or negative. They are finite in numbers. For example, factors of 7 are 1 and 7. Factors of 8 are 1, 2, 4, and 8.

How to Find Number of Factors?

We can find the number of factors of a given number using the following steps.

• Find its prime factorization, i.e. express it as the product of primes.
• Write the prime factorization in the exponent form.
• Add 1 to each of the exponents.
• Multiply all the resultant numbers.
• This product would give the number of factors of the given number.

What Is Prime Factorization?

The prime factorization of a number is writing it as the product of two or more prime numbers. For example the prime factorization of 60 = 2² × 3 × 5.

What Is a Factors Formula?

The factors formula for a number gives the total number of factors of a number. For a number N, whose prime factorization is X^a × Y^b × Z^c, (a+1) (b+1) (c+1) is the total number of factors.

How Do You Factor Equations?

We cannot actually factor equations, but we can factor expressions. Factoring an expression is writing it as the product of two or more expressions. For example: 3x² + 6x = 3x(x + 2)

What Are the Common Factors of 4 and 12?

Factors of 4 = 1, 2 and 4. Factors of 12 = 1, 2, 3, 4 , 6 and 12. Thus, common factors of 4 and 12 are 1, 2, and 4.

What Is a Factor of Every Number?

A factor of a number is a number that is totally divisible by that number. 1 divides every number, thus,1 is a factor of every number.

What Are the Prime Factors of a Number?

The prime factor of a number is the factor of the given number which is a prime number. Factors are the numbers that multiply together to give another number.

Can 0 be a Factor of Any Number?

Since n/0 is undefined for any number, other than zero. Therefore, zero is not a factor of any non-zero number. Other than zero, all integers are factors of zero because 0n = 0, for all numbers.