**What is a Divisor?**

A divisor is a number that divides another number in a division equation. It doesn’t matter whether the number divides it completely or produces a remainder. However, when someone asks you the divisors of a certain number, they are usually referring to **factors. **Both words are used interchangeably, although they have a slight difference. Factors are the numbers that divide a number completely without a remainder.** **Factors can also be defined as all the divisors of a number that do not produce a remainder. Not all divisors are factors of a number, but all factors of a number are divisors of that number.

**For example**

**20 ****÷ 4 = 5**

In the example above, 4 is a divisor of 20. If we replace 4 with 3, it will be a divisor but with a remainder. However, 4 is a factor of 20 while 3 is not. Let’s look at different ways to find the divisors of a number.

**How to find divisors of a number?**

There are several ways to find the divisors of a number. The first option is to divide it by every possible number which is less or equal. But this can be an exhaustive process, especially if the number is very large. If you are working with a small number, then you can use the exhaustive division method.

**Method 1: Using the division Method**

Start by dividing the number by itself until and by all the numbers less than it up to 1.

Next, List only the numbers that did not result in a decimal or remainder and ignore the rest

For example, you can find the divisors of 8 by dividing it with every number less or equal to it.

**8** **÷ 8 = 1**

**8 ÷ 7 = 1.1429**

**8 ÷ 6 = 1.3333**

**8 ÷ 5 = 1.6 **

**8 ÷ 4 = 2**

**8 ÷ 3 = 2.6667**

**8 ÷ 2 = 4**

**8 ÷ 1 = 8**

**Therefore the divisors of 8 are ****1, 2, 4, and 8**

**Method 2: Prime Factorization method**

In this method, you need to list the prime factors of the number first.

For Example, Let’s find the divisors of 24 using the decomposing method

**2 * 2 * 2 * 3**

We already know two divisors which are 1 and 24

From the above list of prime factors, we 2 and 3 are our next divisors.

So far, the divisors are 1, 2, 3, and 24

Now find every possible product from the prime factors of 24

**2 * 2 = 4**

**2* 3 = 6 **

**2* 2 * 2 = 8**

**2* 2 *3 = 12**

**Divisor Facts**

- The divisor can divide a number completely (Without a decimal) or partially (With a decimal).
- A divisor can be a negative or positive number
- The divisor is always larger than the remainder
- Both
**-1**and**1**can divide every number - If there is no remainder, then it means the divisor has divided the dividend completely
- A divisor can also be a number that divides a number without resulting in a remainder.
- When the
**divisor**and the**dividend**are equal, then the quotient will always be**1.**For example,**50****÷****50****= 1** - When the dividend is equal to the quotient, then the divisor is
**1.**For example**50****÷****1****= 50**

**Different ways to represent the divisor**

**How to find the total number of Divisors**

**d(n) = (a+1) (b+1) (C+1)**

Where **a, b, c** are exponents and **n** the number (d stands for divisor in this equation)

**Example 1:** Find the sum of divisors of **72**

**Step 1:** Find the exponents of 72 by breaking it down to its prime factorization

**2 × 2 × 2 × 3 × 3 = 2 ^{3} × 3^{2}**

The exponents of **72** are **3** and **2**

**Step 2:** Apply the exponents to the equation

**d(n) = (a+1) (b+1) (C+1)**

**d(72) = (3+1) (2+1) **

**Step 3: **Simplify the Equation

**d(72) = (4) (3)**

**d(72) = 12**

Therefore the total number of divisors for 72 is** 1**

**Divisors tutorial video**

**Divisor Examples**

**Frequently Asked Questions**

**1. What is the difference between Divisors and Factors? **

Factors are divisors, but divisors are not factors. A divisor is a number that can divide another number with or without a remainder. While a factor is a number that divides another number without a remainder. When asked to list the divisors of a number you can list the factors, and your answer will be correct because all factors are divisors of a number.

**2. What is the formula to find a Divisor?**

There are two ways to find a divisor. If there is a remainder then use this formula **Divisor = (Dividend –Remainder) ÷ Quotient. **In case there is no remainder then the formula will be** Divisor = Dividend ÷ Quotient**

**3. Do Fractions have a Divisor?**

Yes every denominator in a fraction is considered as the divisor. When simplifying the fraction you have to divide the numerator by the denominator

**4. Is an integer considered a divisor of itself? **

Yes every integer divided by itself is one. So an integer is a divisor of itself which does not result into a decimal or remainder.

**5. What is the difference between Divisors and Prime Factors?**

Prime factors are the factors that are prime numbers while divisors are the numbers that can divide an integer with or without a remainder. :