# 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 numberÂ is said to beÂ divisibleÂ if it isÂ entirelyÂ divided by any otherÂ numberÂ without leaving aÂ remainder. When we say that a certain number is completely divisibleÂ by another number, it means that aÂ quotientÂ is aÂ wholeÂ number, and there is noÂ remainder left.

Figure 1 – 15 is divisible by 3 which gives 5

If we want to know theÂ factors of a particularÂ number,Â divisibilityÂ 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 methodsÂ to find theÂ divisibility of a number are by using divisibilityÂ rules,Â prime factorization, and greatestÂ commonÂ divisorÂ (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.

Figure 2 – Prime factorization of 35

## 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 numberÂ 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 divisibilityÂ by 3, we have toÂ addÂ all the numbers of the givenÂ numericalÂ value. If theÂ sumÂ gives aÂ numberÂ that isÂ completelyÂ divided by 3, then the number as aÂ wholeÂ isÂ divisible by 3.

Suppose theÂ numericalÂ value isÂ 35682; add all theÂ numbersÂ of this value 3+5+6+8+2. The answer to thisÂ additionÂ is 24, which is completelyÂ divisibleÂ by 3 so 35682 is also divisible by 3. On theÂ contrary, 35681 is notÂ divisibleÂ by 3 because itsÂ sumÂ is 23, which is notÂ divisibleÂ by 3.

### Divisibility Rule of 4

TheÂ DivisibilityÂ rule of 4 is that if the lastÂ twoÂ digits of anyÂ numericalÂ value areÂ divisibleÂ by 4 and giveÂ zeroÂ remainders, then theÂ numberÂ as a whole isÂ divided by 4.

Let usÂ considerÂ theÂ numericalÂ 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 divisibleÂ by 4 because 4 is not aÂ factorÂ of the last twoÂ digits.

### Divisibility Rule of 5

TheÂ divisibilityÂ rule of 5 is veryÂ simpleÂ and easy toÂ practice. The only numbersÂ divisible by 5 are the ones whose ending value is either a zeroÂ orÂ 5.

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

### Divisibility Rule of 6

A number isÂ completely divisible by 6, leavingÂ zero remainder when it is aÂ factorÂ 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 valuesÂ is aÂ factorÂ of 3, then theÂ number as a whole is divided by 6.

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

### Divisibility Rule of 7

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

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

If again you want toÂ checkÂ theÂ divisibilityÂ of 1435, repeat theÂ processÂ byÂ multiplying 5 by 2 and deductingÂ it from theÂ remaining value, which gives 133, which is easily divisibleÂ by 7. Hence 7 is aÂ factorÂ of the number 14497.

### Divisibility Rule of 8

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

### Divisibility Rule of 9

The rule forÂ divisibilityÂ by 9 isÂ identicalÂ to the rule forÂ divisibilityÂ by 3. Let usÂ understandÂ it by an example.Â 23454Â is aÂ numericalÂ value that we want toÂ check. TheÂ sumÂ of its digits isÂ 18, and 9 is aÂ factorÂ of 18, so 9 is also aÂ factorÂ of 23454.

### Divisibility Rule of 10

Any number that ends at zero isÂ divisibleÂ by 10. ForÂ supposeÂ 25430 and 4950 areÂ divisibleÂ by 10, but 495 and 2543 are notÂ completelyÂ 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.
• 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 sumsÂ producesÂ zero, which satisfies the conditionÂ forÂ divisibility by 11. Hence the numberÂ 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.

Figure 3 – This is how the greatest common divisor is calculated

## 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 divisibleÂ by 5 because its endingÂ numberÂ is neither 5 nor 0.

To checkÂ divisibilityÂ by 7, we have to follow the steps of theÂ divisibilityÂ rule of 7.

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

NowÂ subtractÂ 12 from the rest of theÂ number, i.e., 9345.

9345-12=9333

To check that 9333 isÂ divisibleÂ by 7, repeat the process.

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

SubtractÂ 6 from 933, which gives 927.

927 isÂ divisibleÂ by 7, so 9333 is divisible by 7, and hence theÂ numberÂ 93456 is also divisible by 7.

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