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# Divisibility Rules

A divisor of a number is an integer that divides a number entirely and without leaving any fractional residue. Multiple divisibility criteria may be used to the same number. This allows for the determination of the number's prime factorization to be completed rapidly.

## Introduction

Divisibility rules in mathematics are a collection of precise rules that are applied to a number in order to determine whether or not the provided number is divisible by a specified integer. Some of the most well-known divisibility tests are for the numbers 2 through 20. It enables us to determine the factors and multiples of numbers without having to conduct lengthy division on the values in question. Applying divisibility principles to a number allows a person to mentally determine if a number is divisible by another number or not.

## Body

### Divisibility rules

The procedures outlined below turn a given number into an usually smaller number while maintaining divisibility by the divisor of interest throughout the transformation. As a consequence, unless otherwise specified, the resultant number should be assessed for divisibility using the same divisor as the original number. When it comes to certain processes, it is possible to repeat until the divisibility is clear; nevertheless, when it comes to others (such as when it comes to studying the final n digits), the result must be inspected in other ways.

When there are many rules for a divisor, the rules are often organised such that the rules that are acceptable for numbers with many digits come first, followed by the rules that are beneficial for numbers with fewer digits.

## Divisibility by 2

To begin, choose any number (in this case, 376 would suffice) and make a note of the last digit of the number, disregarding the other digits. Then, taking that digit (6) and disregarding the remainder of the number, decide whether or not it is divisible by two. It follows that if the new number is divisible by 2, then the old number is also divisible by 2.

Divisibility by three or nine

In order to get the answer, start with any number (in this case, 492) and add up each digit in the number (4 + 9 + 2 = 15). Then, using that total (15), determine whether or not it is divisible by three. Only if and only if the sum of the original number’s digits is divisible by 3 (or 9) is the original number considered divisible by 3 (or 9). (or 9).

If you add the digits of a number together, and then repeat the procedure with the result until only one digit is left, you will get the remainder of the original number if the number is divided by nine, according to this formula (unless that single digit is nine itself, in which case the number is divisible by nine and the remainder is zero).

## Divisibility by 4

As a general rule, if the number formed by the last two digits of a number is divisible by 4, the original number is also divisible by 4. This is because 100 is divisible by 4, and so adding hundreds, thousands, and other large numbers is simply adding another number that is divisible by four; however, this is not always the case. If any number finishes in a two-digit number that you know is divisible by four (e.g., 24, 04, 08, etc.), then the whole number will be divisible by four, regardless of what comes before the final two digits of the last two-digit number.

An alternative method would be to simply divide a number by two and then verify the result to see whether the number is divisible by two. If this is the case, the original number is divisible by four times. The result of this test, in addition, is the same as the result of the original number divided by 4.

## Divisibility by 5

It is simple to discover if a number (475) is divisible by 5 by looking at the final digit and seeing whether it is either a zero or a five. If the final number is either 0 or 5, the whole number is divisible by 5. Otherwise, the full number is divisible by 10.

If the final digit of the number is zero, the outcome will be the sum of the other digits multiplied by two, as shown in the example. For example, the number 40 ends in a zero, so take the remaining digits (4) and multiply them by two (4 x 2 = 8), and you have the answer. The answer is the same as the result of dividing 40 by 5 (40/5 = 8), which is also 8.

Suppose the final digit of a number is 5, then the result is the sum of the previous digits multiplied by two plus one (i.e., 5). When a number ends in a 5, for example, take the remaining digits (12), multiply them by two (12 2 = 24), and then add one (24 + 1 = 25) to get the final result. The outcome is the same as the result of dividing 125 by 5 (125/5=25), which is equal to 25.

## Divisibility by 6

Checking the original number to verify whether it is both an even number (divisible by 2) and a multiple of three (divisible by three) determines its divisibility by six. This is the most effective test to employ.

Assuming the number is divisible by six, begin by multiplying it by two and dividing the result by two (246 divided by two is 123). Then divide the result by three (123 /3 = 41) to get the final answer. This is the same as the original number split by six (246 / 6 = 41), which is the same as the answer. divisibility rules and examples are the best way to learn divisibility rules.

## Conclusion

A divisibility rule is a kind of shortcut that allows us to determine whether or not a given integer is divisible by a divisor by analysing its digits, rather than having to go through the whole division procedure. Multiple divisibility criteria may be used to the same number, which allows for the determination of the number’s prime factorization to be completed rapidly. A divisor of a number is an integer that divides a number entirely and without leaving any fractional residue.  The digits of a number may be used to find the true divisor of a number using specific principles that we will discuss in this section. The rules governing divisibility are referred to as divisibility rules.

## Frequently asked questions

Get answers to the most common queries related to the JEE Examination Preparation.

## What criteria do you use to determine divisibility?

Ans. When a number can be split evenly by another number, it is said to be divisible by that number; that is, when a...Read full

## What is the law of divisibility for the numbers 7 and 11?

Ans. Divisibility by the numbers 7 and 11 is determined by taking the final digit of the number, doubling it, and th...Read full

## When it comes to the number 125, what is the rule of divisibility?

Ans: In order to be divided by 125, the number generated by the hundreds, tens, and units digits of a given integer ...Read full

## Is the number 12 divisible by three, yes or no?

Ans. Our division yielded a whole integer, therefore we know that 12 is divisible by three.

## What is the smallest number that is divisible by six?

Ans: Natural numbers are divisible by 6 if and only if they are divisible by two and three as well as six. It is nec...Read full