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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?

AÂ **number**Â is said to beÂ **divisible**Â if it isÂ **entirely**Â divided by any otherÂ **number**Â without leaving aÂ **remainder**. When we say that a certain **number** is completely **divisible**Â by another number, it means that aÂ **quotient**Â is aÂ **whole**Â number, and there is noÂ **remainder** left.

If we want to know theÂ **factors** of a **particular**Â number,Â **divisibility**Â 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 zeroÂ **remainders**.

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

## Prime Factorization

One of the **methods** for **determining** divisibility is by using prime **factorization**.Â 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**Â isÂ **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 easyÂ **method** to figure out if an integer is **dividable **by 3. For **divisibility**Â by 3, we have toÂ **add**Â all the numbers of the givenÂ **numerical**Â value. If theÂ **sum**Â gives aÂ **number**Â that isÂ **completely**Â divided by 3, then the number as aÂ **whole**Â isÂ **divisible** by 3.

Suppose theÂ **numerical**Â value isÂ **35682**; add all theÂ **numbers**Â of this value 3+5+6+8+2. The answer to thisÂ **addition**Â is 24, which is completelyÂ **divisible**Â by 3 so 35682 is also divisible by 3. On theÂ **contrary**, 35681 is notÂ **divisible**Â by 3 because itsÂ **sum**Â is 23, which is notÂ **divisible**Â by 3.

### Divisibility Rule of 4

TheÂ **Divisibility**Â rule of 4 is that if the lastÂ **two**Â digits of anyÂ **numerical**Â value areÂ **divisible**Â by 4 and giveÂ **zero**Â remainders, then theÂ **number**Â as a whole isÂ **divided** by 4.

Let usÂ **consider**Â theÂ **numerical**Â value 7512; the last twoÂ **digits**, 12, areÂ **divisible** by 4, 12/4=3, so 7512 is divisible by 4. But **7502** is not completely **divisible**Â by 4 because 4 is not aÂ **factor**Â of the last twoÂ **digits**.

### Divisibility Rule of 5

TheÂ **divisibility**Â rule of 5 is veryÂ **simple**Â and easy toÂ **practice**. The only **numbers**Â **divisible** by 5 are the ones whose **ending** value is either a **zero**Â orÂ **5**.

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

### Divisibility Rule of 6

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

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

### Divisibility Rule of 7

TheÂ **divisibility**Â rule of 7 is a bitÂ **complicated** as compared to other **numbers’** divisibility. Let us **understand**Â this by using anÂ **example**.

Suppose a value of 14497; first, we have to **multiply**Â theÂ **last**Â number, i.e., 7, by 2, which gives 14. NowÂ **subtract**Â 14 from the remainingÂ **number**, i.e., 1449, which gives 1435. IfÂ **1435**Â is divisible by 7, thenÂ **14497**Â is alsoÂ **divisible** by 7.

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

### Divisibility Rule of 8

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

### Divisibility Rule of 9

The rule forÂ **divisibility**Â by 9 isÂ **identical**Â to the rule forÂ **divisibility**Â by 3. Let usÂ **understand**Â it by an example.Â **23454**Â is aÂ **numerical**Â value that we want toÂ **check**. TheÂ **sum**Â of its digits isÂ **18**, and 9 is aÂ **factor**Â of 18, so 9 is also aÂ **factor**Â of 23454.

### Divisibility Rule of 10

Any number that ends at zero isÂ **divisible**Â by 10. ForÂ **suppose**Â 25430 and 4950 areÂ **divisible**Â by 10, but 495 and 2543 are notÂ **completely**Â 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 anÂ **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**Â producesÂ **zero**, which satisfies the **condition**Â forÂ **divisibility** by 11. Hence the **number**Â 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 notÂ **divisible**Â by 5 because its endingÂ **number**Â is neither 5 nor 0.

To checkÂ **divisibility**Â by 7, we have to follow the steps of theÂ **divisibility**Â rule of 7.

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

NowÂ **subtract**Â 12 from the rest of theÂ **number**, i.e., 9345.

9345-12=9333

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

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

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

927 isÂ **divisible**Â by 7, so 9333 is divisible by 7, and hence theÂ **number**Â 93456 is also divisible by 7.

*All images are created using GeoGebra. *