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# Divisible|Definition & Meaning

## Definition

When a **number** gets completely **divided** by another **number**, without leaving any **remainder**, that number is said to be **divisible** by the other **number**. For example, **10** is completely **divided** by 2 and thus is **divisible** by **2** but it is not completely **divided** by **3** – leaving a **remainder** of 1 – and so is not **divisible** by 3.

## What Is Meant by Divisible?

A **number** is said to be **divisible** if it is **entirely** divided by any other **number** without leaving a **remainder**. When we say that a certain **number** is completely **divisible** by another number, it means that a **quotient** is a **whole** number, and there is no **remainder** left.

If we want to know the **factors** of a **particular** number, **divisibility** plays an important role in it. For example, 2,3,4, and 6 are the factors of 12 because all of these divide 12 completely, giving zero **remainders**.

Some of the **methods** to find the **divisibility** of a number are by using **divisibility** rules, **prime** factorization, and **greatest** common **divisor** (GCD).

## Prime Factorization

One of the **methods** for **determining** divisibility is by using prime **factorization**. If a number is the **factor** of the prime **factorization** of a particular number, then that particular **number** is divisible by it. For example, take the **number** 35; we know that **5*7=35,** so 5 and 7 are the **prime** factors of 35 and hence 35 is **divisible** by 5 and 7.

## Divisibility Rule

To review whether a particular **number** is completely divisible by 2 or not, we have to look at the last **number** of the given **numerical** value. If a **number** is **even** or it ends with 0,2,4,6 or 8, then it is completely **divisible** by 2.

### Divisibility Rule of 1

We know that every number is **completely** **divided** by 1, leaving no **remainder**. No matter how **large** the number is, dividing it by **1** will yield the number itself as a **quotient**. For example,12876 is **divisible** by 1. So there is no specific **condition** for any number divided by 1.

### Divisibility Rule of 2

To review whether a particular **number** is completely divisible by 2 or not, we have to look at the last **number** of the given **numerical** value. If a **number** is **even** or it ends with 0,2,4,6 or 8, then it is completely **divisible** by 2.

For example, 3256 is **divisible** by 2 because it ends with an **even** number, but if we consider the **number** 3257, whose **last** digit is 7, thus not being an **even** number hence 3257 is not **divisible** by 2.

### Divisibility Rule of 3

There is an easy **method** to figure out if an integer is **dividable **by 3. For **divisibility** by 3, we have to **add** all the numbers of the given **numerical** value. If the **sum** gives a **number** that is **completely** divided by 3, then the number as a **whole** is **divisible** by 3.

Suppose the **numerical** value is **35682**; add all the **numbers** of this value 3+5+6+8+2. The answer to this **addition** is 24, which is completely **divisible** by 3 so 35682 is also divisible by 3. On the **contrary**, 35681 is not **divisible** by 3 because its **sum** is 23, which is not **divisible** by 3.

### Divisibility Rule of 4

The **Divisibility** rule of 4 is that if the last **two** digits of any **numerical** value are **divisible** by 4 and give **zero** remainders, then the **number** as a whole is **divided** by 4.

Let us **consider** the **numerical** value 7512; the last two **digits**, 12, are **divisible** by 4, 12/4=3, so 7512 is divisible by 4. But **7502** is not completely **divisible** by 4 because 4 is not a **factor** of the last two **digits**.

### Divisibility Rule of 5

The **divisibility** rule of 5 is very **simple** and easy to **practice**. The only **numbers** **divisible** by 5 are the ones whose **ending** value is either a **zero** or **5**.

For example, **43265** is **divisible** by 5 because it ends with 5, but 43266 is not **divisible** by 5 due to the **ending** number, i.e., 6.

### Divisibility Rule of 6

A number is **completely** divisible by 6, **leaving** zero **remainder** when it is a **factor** of both 2 and 3. In other words, we can say that if a **number** ends with even **values** and the **summation** of all of its **values** is a **factor** of 3, then the **number** as a whole is **divided** by 6.

For example, 540 is **divisible** by 6 as it **ends** with an **even** number(hence 2 is a factor), and adding its **number** gives 9, which is **divisible** by 3 (hence 540 is divisible by 3).

### Divisibility Rule of 7

The **divisibility** rule of 7 is a bit **complicated** as compared to other **numbers’** divisibility. Let us **understand** this by using an **example**.

Suppose a value of 14497; first, we have to **multiply** the **last** number, i.e., 7, by 2, which gives 14. Now **subtract** 14 from the remaining **number**, i.e., 1449, which gives 1435. If **1435** is divisible by 7, then **14497** is also **divisible** by 7.

If again you want to **check** the **divisibility** of 1435, repeat the **process** by **multiplying** 5 by 2 and **deducting** it from the **remaining** value, which gives 133, which is easily **divisible** by 7. Hence 7 is a **factor** of the number 14497.

### Divisibility Rule of 8

An integer is completely **divisible** by 8 if and only if the leftmost **three** values of that number are **divided** by 8, **leaving** zero remainders. For instance, **35408** is a number, and its last **three** digits, i.e., 408, are **divided** by 8 **completely**, so the **number** as a whole is **divisible** by 8.

### Divisibility Rule of 9

The rule for **divisibility** by 9 is **identical** to the rule for **divisibility** by 3. Let us **understand** it by an example. **23454** is a **numerical** value that we want to **check**. The **sum** of its digits is **18**, and 9 is a **factor** of 18, so 9 is also a **factor** of 23454.

### Divisibility Rule of 10

Any number that ends at zero is **divisible** by 10. For **suppose** 25430 and 4950 are **divisible** by 10, but 495 and 2543 are not **completely** divisible by 10.

### Divisibility Rule of 11

For an integer to be completely dividable by 11, we have to follow a few steps.

**Observe**the number carefully.**Add**the digits alternatively.**Subtract**both the sums obtained in the above step.- If the difference is
**zero**or a**number**that is**multiple**of 11, then the number as a whole is**divisible**by 11.

Let’s apply these steps to an **example.** Consider the number 723954. Adding the alternate numbers, i.e., 7+3+5 and 2+9+4. The sums are 15 and 15. **Subtracting** both **sums** produces **zero**, which satisfies the **condition** for **divisibility** by 11. Hence the **number** 723954 is divisible by 11.

### Greatest Common Divisors (GCD)

**Divisibility** can also be determined by using the Euclidean algorithm. It is a method to locate the GCD of two numbers. We know that GCD gives out the **largest** number providing no **remainder** when dividing both **numbers.**

For example, the **GCD** of 12 and 8 is 4, which means that 4 is the largest number that **divides** both 12 and 8 without leaving a remainder. So 12 and 8 both are divisible by 4.

## Solved Examples of Divisibility of a Number

### Example 1

Check whether 23487 is **divisible** by 3 or not.

### Solution

The first step of the **divisibility** rule of 3 is adding all the values.

2+3+4+8+7=24

Since 24 is **divisible** by 3, so the number 23487 is also **divisible** by 3.

### Example 2

Check whether 93456 is **divisible** by 5 or 7.

### Solution

The number 93456 is not **divisible** by 5 because its ending **number** is neither 5 nor 0.

To check **divisibility** by 7, we have to follow the steps of the **divisibility** rule of 7.

**Multiplying** the last digit by 2, 6*2=12.

Now **subtract** 12 from the rest of the **number**, i.e., 9345.

9345-12=9333

To check that 9333 is **divisible** by 7, repeat the process.

The last digit, 3, **multiplied** by 2, gives 6.

**Subtract** 6 from 933, which gives 927.

927 is **divisible** by 7, so 9333 is divisible by 7, and hence the **number** 93456 is also divisible by 7.

*All images are created using GeoGebra. *