The document discusses tests of divisibility to determine if a number is perfectly divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12 without remainder. It provides the specific tests, which involve examining digits in certain places of the number or adding all digits together. The tests are then demonstrated with examples of numbers that are and are not divisible by each number.

1. Numbers-I: Divisibility This presentation will tell you how to find whether a number – any number – can be perfectly divided 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 no remainder left at the end.)

2. Numbers-I: Divisibility The methods used to find whether a given number is divisible by another number are called Tests of Divisibility . This presentation will show the Tests of Divisibility for 2, 3, 4, 5, 6, 7, 9 10, 11 and 12.

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

4. Numbers-I: Divisibility Test of Divisibility for 2 To know whether a number is divisible by 2, simply look at its unit digit. If the unit digit has 0, 2, 4, 6, or 8 , the number will be divisible by 2, else it will not be divisible by 2. Hence 27 8 , 99 6 and 405 0 are divisible by 2, while 66 3 , 868 9 and 1034 3 are not divisible by 2.

5. Numbers-I: Divisibility Test of Divisibility for 4 To know whether a number is divisible by 4, simply look at its tens and unit digit. If these last two digits form a number divisible by 4, the entire number is divisible by 4. Hence 14 26 , 60 43 and 19 98 are not divisible by 4, while 2 64 , 40 48 , 3 36 , and 9 72 are divisible by 4 (because 64, 48, 36 and 72 are divisible by 4) .

6. Numbers-I: Divisibility Test of Divisibility for 8 To know whether a number is divisible by 8, look at the number formed by the hundreds, tens and units digit in that order. If the number is divisible by 8, the entire number will be divisible by 8, else it will not be divisible by 8. Hence 57 662 , 304 588 and 671 690 are not divisible by 8, while 986 104 , 4031 416 and 788 208 are divisible by 8 (because 154, 412 and 208 are divisible by 8).

7. Numbers-I: Divisibility Test of Divisibility for 3 To know whether a number is divisible by 3, add ALL the digits of the number. If the total is divisible by 3, the number is divisible by 3. e.g. 12456 => 1 + 2 + 3 + 4 + 5 + 6 = 18 . We know 18 is divisible by 3, hence 12456 is divisible by 3 . 14213=> 1 + 4 + 2 + 1 + 3 = 11 . We know 11 is not divisible by 3, hence 14213 is not divisible by 3 .

8. Numbers-I: Divisibility Test of Divisibility for 6 If a number is divisible by 6, it must satisfy both the following conditions: 1) It should be even and 2) It should be divisible by 3. e.g. 12456 => 1 + 2 + 3 + 4 + 5 + 6 = 18 . It is divisible by 3 and is even. Hence 12456 is divisible by 6 . e.g. 30129 is not even, hence 30129 not divisible by 6 . e.g. 2116 => 2 + 1 + 1 + 6 = 10 . We know 10 is not divisible by 3, hence 2116 is not divisible by 6 .

9. Numbers-I: Divisibility Test of Divisibility for 9 To know whether a number is divisible by 9, add ALL the digits of the number. If the total is divisible by 9, the number is divisible by 9. e.g. 12456 => 1 + 2 + 3 + 4 + 5 + 6 = 18 . We know 18 is divisible by 9, hence 12456 is divisible by 9 . 14219=> 1 + 4 + 2 + 1 + 9 = 17 . We know 17 is not divisible by 9, hence 14219 is not divisible by 9 .

10. Numbers-I: Divisibility Test of Divisibility for 10 To know whether a number is divisible by 10, simply look at its unit digit. If the unit digit has 0 the number will be divisible by 10, else it will not be divisible by 10. Hence 27 0 , 99 0 and 405 0 are divisible by 10, while 66 3 , 868 9 and 1034 3 are not divisible by 10.

11. Numbers-I: Divisibility Test of Divisibility for 12 If a number is divisible by 12, it must satisfy both the following conditions: 1) It should be divisible by 4 and 2) It should be divisible by 3. e.g. 12456 => 1 + 2 + 3 + 4 + 5 + 6 = 18 . It is divisible by 3. The last two digits 124 56 are divisible by 4. Hence the number is divisible by 12. e.g. 2116 => 2 + 1 + 1 + 6 = 10 . We know 10 is not divisible by 3, hence 2116 is not divisible by 12 .

12. Numbers-I: Divisibility Test of Divisibility for 11 Consider the number 6138. Now consider the digits in alternate places: 6 1 3 8 . Add the first set of alternate digits 6 + 3 = 9 . Add the second set of alternate digits 1 + 8 = 9. If the difference in the two totals is 0, 11, 22, 33, …, the number is divisible by 11 . Here the difference is 0 (= 9 - 9) , hence 6138 is divisible by 11. By the same rule 7 5 4 6 9 1 3 is also divisible by 11. 