# Divisibility Tests

SupposeÂ you’re at a stationary and you need to find which deal is better by using divisibility rules. Let’s say 2 pencils cost Rs 6 and in another store, 4 pencils cost Rs 8. Which deal is better? We know that 6 is divisible by 2, so each pencil costs Rs 3. Also, 8 is divisible by 4 which means each pencil cost Rs 2. We know that the shop with 4 pencils that cost Rs 8 is the best deal. Let us study the divisibility rules in detail.

## Divisibility Tests

You have bought 1235 chocolates and want to distribute them in your class. How will you know with what number to divide them without actually performing the operation of division? This is where the concept of divisibility rules comes into play.

The process of whether the given number is exactly divisible by another number without actually performing the operation of division is called the test of divisibility. Being exactly divisible means that on division there is no remainder left.Â  For understanding the divisibility rules for any number we need to know its divisibility with numbers like 2,3,4,5,9,10 and 11.

## Divisibility Rules

### Divisibility Test by 2

If theÂ ones digit or unit’s place digitÂ of a number is either 0, 2, 4, 6 0r 8Â Â then the number is said to be divisibleÂ by 2. In other terms, if the last digit of the number is even then it is always divisible by 2.Â For example, let’s takeÂ  24. The last digit is even. 4 is divisibleÂ by 2, so the number is divisible by 2.

### Divisibility Test by 4

AÂ number is divisible byÂ 4Â  if the number formed by the last two digits isÂ divisible byÂ 4. For example, let’s take 12343684. You don’t need to worry about all the numbers, just check the last 2 digits of the numbers. If the last two digits i.e 84 are divisibleÂ by 4than the number is divisible by 4.

### Divisibility Test by 3 and 9

A number is divisible by 3 if theÂ sum of its digitsÂ is divisible byÂ 3. Let’s have a look at some examples.

• 121: Here the sum of the digit is 4. Is 4=the number 4 divisible by 3? No.
• 123:Â  Here the sum of the digit is 6 and yes 6 is divisible by 3. So the number 123 is divisible by 3.

Same rules are applied to the divisibilityÂ test of 9. A number is divisible by 9 if theÂ sum of its digitsÂ is divisible by 9.

### Test of Divisibility by 5

A number is exactly divisible by 5 if it has the digits 0 or 5 at one’s place. The numbers like 15,120,205,4400 etc. are exactly divisible by 5 as these numbers have 0 or 5 in the one’s place. Let’s look at an example.

In the numbers 21345650 and 459022, which one is exactly divisible by 5? Out of both the numbersÂ  21345650 and 459022, the first number i.e 21345650 is exactly divisible by 5 as the ones place is occupied by 0.

### Test of Divisibility by 6

A number is exactly divisible by 6 if that number is divisible by 2 and 3 both. For checking the divisibility of the number with 6 we have to apply the rule of divisibility tests that we apply for numbers 2 and 3. Let’s look at an example. Using the divisibility tests rule for 6, check whether the following number is divisible by 6: 12,581.

In the number 12,581 the ones digit does not have any of the numbers like 0,2,4,6,8 in the ones place. So the number is not divisible by 2. Hence the number is not divisible by 6 because for a number to be divisible by 6, it has to be divisible by 2 and 3 both.

### Test of Divisibility by 8

The test of divisibility by 8 is used in numbers with four or more digits. A number is exactly divisible by 8 if the number formed by the digits in ones, tens and hundredth place is divisible by 8. In the number 5864, we see that the number formed by the digits in ones, tens and hundredth place, 864 which is divisible by 8, so the number is divisible by 8.

For example, whether the following number is divisible by 8: 5,87,824. In the numbers given above, the number 5,87,824 has 824 in the ones, tens and hundredth place. This number is exactly divisible by 8, hence the number is divisible by 8.

### Divisibility Test by 7

Test 1

Take the digits of the number in reverse order, from right to left, multiplying them successively by the digits 1, 3, 2, 6, 4, 5, repeating with this sequence of multipliers as long as necessary. Add the products. If the sum is divisible by 7Â then it passes the divisibility test of 7.

Example:Â  Is the number 1603 divisible by 7?
Solution: Mutiplying by the respective digits, 3Ã— 1 + 0Â Ã— 3 + 6Â Ã— 2 + 1Â Ã— 6= 21. This number is divisble by 7.

Test 2

Double the last digit and subtract it from the remaining leading truncated number. If the result is divisible byÂ 7, then so was the original number. Apply this rule over and over again as necessary.

Example:Â  Is the number 1603 divisible by 7?
Solution: Applying the method, 160 – 2Ã— 6 = 154. This number is divisible by 7.

### Divisibility Test by 11

Subtract the last digit from the remaining leading truncated number. If the result is divisible by 11Â then so was the first number. Apply this rule over and over again as necessary. For example, let’s take the number 19151.

19151 â†’ 1915 – 1 = 1914
1914 â†’ 191 – 4 = 187
187 â†’ 18 – 7 = 11
Hence, 19151 is divisible by 11.

### Divisibility Rules for Higher Numbers

• A number is exactly divisible by 10 if the one’s place of the number is occupied by 0. So, the numbers with 0 in the one’s place are divisible by 10.
• A number is divisible by 13 if the number obtained by subtractingÂ  9 times the last digit from the remaining digits of the number is divisible by 13.
• A number is divisible byÂ 17 if the number obtained after subtractingÂ 5 times the last digit from the rest is divisible byÂ 17.

### Solved Examples for You

Question 1: Which of the following numbers is divisible byÂ 14?

1. 4683
2. 7321
3. 1428
4. 5631

Answer: C is the correct option. The divisibility rule for 14 is that if the number is divisible by bothÂ 2 and 7 then the number is exactly divisible by 14. Here the last digit is, even so, the number 1428 is divisible by 2.Â 1428 is multiple of 7 so this number is also divisible by 7.

Question 2:Â Explain the divisibility rule of 2?

Answer:The divisibility rule for two tells us that any number whose last digit happens to be 0, 2, 4, 6, or 8 shall be divisible by 2. In other words, an even number is able to be divisible by 2. Also, an odd number will not be divisible by two.

Question 3:Â Explain the divisibility rule for 11?

Answer:Â The divisibility rule of eleven tells us that one must subtract and afterwards add the digits in an alternating pattern. This alternating pattern is from left to right. In case the answer is 0 or 11, then the result shall be divisible by 11.

Question 4:Â What is the importance of divisibility rules?

Answer:Â Divisibility rules are very useful as they help to quickly determine if a number can be divided by 2, 3, 4, 5, 9, and 10 without undertaking long division.

Question 5:Â Explain the divisibility rule of 5?

Answer:Â A number that is divisible by 5 should end in either 0 or 5. For this rule, letâ€™s take a look at the last digit of 34,780. The last digit is a 0 which makes this number is divisible by 5.

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