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Numbers-I: DivisibilityNumbers-I: DivisibilityThis presentation will tell you how to find whether This presentation will tell you how to find whether a number – any number – can be a number – any number – can be perfectlyperfectly divided divided by the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12.by the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12.

(Note: When we say “perfectly divided by” we mean there will be (Note: When we say “perfectly divided by” we mean there will be no remainder left at the end.)no remainder left at the end.)


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Numbers-I: DivisibilityNumbers-I: DivisibilityThe methods used to find whether a given number The methods used to find whether a given number is divisible by another number are called is divisible by another number are called Tests of Tests of DivisibilityDivisibility..

This presentation will show the Tests of This presentation will show the Tests of Divisibility for Divisibility for 2, 3, 4, 5, 6, 7, 9 10, 11 2, 3, 4, 5, 6, 7, 9 10, 11 andand 12. 12.


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Numbers-I: DivisibilityNumbers-I: DivisibilityImportant: Before proceeding further, you must Important: Before proceeding further, you must know what do we mean by units, tens and know what do we mean by units, tens and hundredth digits. hundredth digits. The right-most digit of a number is called Unit Digit. E.g. 7 is the The right-most digit of a number is called Unit Digit. E.g. 7 is the unit digit in 415unit digit in 41577.. The second right-most digit of a number is called the Tens digit of The second right-most digit of a number is called the Tens digit of the number.e.g. in 25the number.e.g. in 25441, 4 is the digit in the tens’ place.1, 4 is the digit in the tens’ place. The third right-most digit of a number if called the Hundreds digit The third right-most digit of a number if called the Hundreds digit of a the number. e.g. in 1of a the number. e.g. in 15539, 5 is the digit in the hundreds’ place.39, 5 is the digit in the hundreds’ place.


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Numbers-I: DivisibilityNumbers-I: DivisibilityTest of Divisibility for 2Test of Divisibility for 2

To know whether a number is divisible by 2, To know whether a number is divisible by 2, simply look at its unit digit. If simply look at its unit digit. If the unit digit has 0, the unit digit has 0, 2, 4, 6, 2, 4, 6, oror 8 8, the number will be divisible by 2, else , the number will be divisible by 2, else it will not be divisible by 2.it will not be divisible by 2.

Hence 27Hence 2788, 99, 9966 and 405 and 40500 areare divisible by 2, while divisible by 2, while 666633, 868, 86899 and 1034 and 103433 are are notnot divisible by 2. divisible by 2.


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To know whether a number is divisible by 4, To know whether a number is divisible by 4, simply look at its tens and unit digit. simply look at its tens and unit digit. If these last If these last two digits form a number divisible by 4, the two digits form a number divisible by 4, the entire number is divisible by 4. entire number is divisible by 4.

Hence 14Hence 142626, 60, 604343 and 19 and 199898 are are notnot divisible by 4, divisible by 4, while 2while 26464, 40, 404848, 3, 33636, and 9, and 97272 areare divisible by 4 divisible by 4 (because 64, 48, 36 and 72 are divisible by 4)(because 64, 48, 36 and 72 are divisible by 4)..


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To know whether a number is divisible by 8, look To know whether a number is divisible by 8, look at the number formed by the hundreds, tens and at the number formed by the hundreds, tens and units digit in that order. If units digit in that order. If the number is divisible the number is divisible by 8, the entire number will be divisible by 8, by 8, the entire number will be divisible by 8, else it will not be divisible by 8.else it will not be divisible by 8.Hence 57Hence 57662662, 304, 304588588 and 671 and 671690690 are not divisible are not divisible by 8, while 986by 8, while 986104104, 4031, 4031416416 and 788 and 788208208 are are divisible by 8 divisible by 8 (because 154, 412 and 208 are divisible by 8).(because 154, 412 and 208 are divisible by 8).


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To know whether a number is divisible by 3, add To know whether a number is divisible by 3, add ALL the digits of the number. If the total is ALL the digits of the number. If the total is divisible by 3, the number is divisible by 3.divisible by 3, the number is divisible by 3.

e.g. 12456 => e.g. 12456 => 1 1 ++ 2 2 ++ 3 3 ++ 4 4 ++ 5 5 ++ 6 6 = = 1818. We know . We know 18 is divisible by 3, 18 is divisible by 3, hence 12456 is divisible by 3hence 12456 is divisible by 3..

14213=>14213=>1 1 ++ 4 4 ++ 2 2 ++ 1 1 ++ 3 3 = = 1111. We know 11 is not . We know 11 is not divisible by 3, divisible by 3, hence 14213 is not divisible by 3hence 14213 is not divisible by 3. .


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If a number is divisible by 6, it must satisfy If a number is divisible by 6, it must satisfy bothboth the the following conditions:following conditions:1) It should be even1) It should be even and 2) It should be divisible by 3.and 2) It should be divisible by 3.

e.g. 12456 => e.g. 12456 => 1 1 ++ 2 2 ++ 3 3 ++ 4 4 ++ 5 5 ++ 6 6 = = 1818. It is divisible by . It is divisible by 3 and is even. 3 and is even. Hence 12456 is divisible by 6Hence 12456 is divisible by 6. .

e.g. 30129 is not even, e.g. 30129 is not even, hence 30129 not divisible by 6hence 30129 not divisible by 6..

e.g. 2116 =>e.g. 2116 =>2 2 ++ 1 1 ++ 1 1 ++ 6 6 = = 1010. We know 10 is not . We know 10 is not divisible by 3, hence divisible by 3, hence 2116 is not divisible by 62116 is not divisible by 6..


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To know whether a number is divisible by 9, add To know whether a number is divisible by 9, add ALL the digits of the number. If the total is ALL the digits of the number. If the total is divisible by 9, the number is divisible by 9.divisible by 9, the number is divisible by 9.

e.g. 12456 => e.g. 12456 => 1 1 ++ 2 2 ++ 3 3 ++ 4 4 ++ 5 5 ++ 6 6 = = 1818. We know . We know 18 is divisible by 9, 18 is divisible by 9, hence 12456 is divisible by 9hence 12456 is divisible by 9..

14219=>14219=>1 1 ++ 4 4 ++ 2 2 ++ 1 1 ++ 9 9 = = 1717. We know 17 is not . We know 17 is not divisible by 9, divisible by 9, hence 14219 is not divisible by 9hence 14219 is not divisible by 9. .


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To know whether a number is divisible by 10, To know whether a number is divisible by 10, simply look at its unit digit. If simply look at its unit digit. If the unit digit has 0 the unit digit has 0 the number will be divisible by 10, else it will not the number will be divisible by 10, else it will not be divisible by 10.be divisible by 10.

Hence 27Hence 2700, 99, 9900 and 405 and 40500 areare divisible by 10, while divisible by 10, while 666633, 868, 86899 and 1034 and 103433 are are notnot divisible by 10. divisible by 10.


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If a number is divisible by 12, it must satisfy If a number is divisible by 12, it must satisfy bothboth the following conditions:the following conditions:1) It should be divisible by 41) It should be divisible by 4 and 2) It should be and 2) It should be divisible by 3.divisible by 3.

e.g. 12456 => e.g. 12456 => 1 1 ++ 2 2 ++ 3 3 ++ 4 4 ++ 5 5 ++ 6 6 = = 1818. It is . It is divisible by 3. The last two digits 124divisible by 3. The last two digits 1245656 are are divisible by 4. Hence the number is divisible by 12.divisible by 4. Hence the number is divisible by 12.

e.g. 2116 =>e.g. 2116 =>22 ++ 11 ++ 11 ++ 66 = = 1010. We know 10 is not . We know 10 is not divisible by 3, hence divisible by 3, hence 2116 is not divisible by 122116 is not divisible by 12..


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Consider the number 6138. Now consider the digits Consider the number 6138. Now consider the digits in alternate places: in alternate places: 66113388. .

Add the first set of alternate digits Add the first set of alternate digits 66 + + 33 = = 99..

Add the second set of alternate digits Add the second set of alternate digits 11 + + 88 = = 9. 9.

If the difference in the two totals is 0, 11, 22, 33, If the difference in the two totals is 0, 11, 22, 33, …, the number is divisible by 11…, the number is divisible by 11. . Here the difference is 0 Here the difference is 0 (= 9 - 9)(= 9 - 9), hence 6138 is divisible by 11. By the same rule , hence 6138 is divisible by 11. By the same rule 77554466991133 is is also divisible by 11.also divisible by 11.

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