 Hello and welcome to the session. In this session we will discuss tests of divisibility. We already know how to check the divisibility by the numbers 10, 5, 2, 9 and 3. Now the reasons for the divisibility of the numbers by 10, 5, 2, 9 or 3 can be given when we write the numbers in general form. First let's check out divisibility by 10. We know that a number is divisible by 10 when its once digit is 0. Let's see how this works. Let's consider this number CVA. Now this can be written as 100 C plus 10 B plus A. Here we have A is the once digit, B is the 10 digit and C is the 100 digit and so on. And these dots that we have shown, this shows that there may be more digits to the left of C. Now since we know that 100 are all divisible by 10, so we can say that 10 B, 100 C and so on would also be divisible by 10. And as far as this number A is concerned, if the given number is divisible by 10, then we say that A is divisible by 10 and this would be possible only when we have A equal to 0. This is how the divisibility test of a number by 10 works. Let's consider the number 120. This can be written as 100 into 1 plus 10 into 2 plus 0. That is we have 120 is equal to 100 plus 20 plus 0. Now you know that 120 are both divisible by 10 and this number is 0. Now we say this number 120 is divisible by 10. Next we have divisibility by 5. The once digit of a number is 0 or 5, then it is divisible by 5. Now consider any number like C B A. This can be written as 100 C plus 10 B plus A. These dots are to show that there can be more digits to the left of C. We know that the numbers 100 are divisible by 10 and thus the numbers 10 B 100 C and so on would be divisible by 10 and they would hence be divisible by 5 since we have like 10 is equal to 2 into 5. And for this number A to be divisible by 5 that is A is divisible by 5 if the number is divisible by 5. So we have A has to be either 0 or 5. Now let's consider the number 120 itself. This is written as 100 into 1 plus 10 into 2 plus 0. That is we have 120 is equal to 100 plus 20 plus 0. These numbers 120 are divisible by 10 and so we say that they are divisible by 5 also. And as you can see this number is 0 thus we can say that 120 is divisible by 5. Next is divisibility by the number 2. We know that if the ones digit of a number is 0, 2, 4, 6 or 8 then we say the number is divisible by 2. Consider the number CBA this is written as 100 C plus 10 B plus A. Since we have that 10 100 are divisible by 2 thus the numbers 100 C 10 B would be divisible by 2. And this number A is divisible by 2 if the given number is divisible by 2. And this would be possible only when we have A equal to 0, 2, 4, 6 or 8. Consider this number 124 this is written as 100 into 1 plus 10 into 2 plus 1 into 4. That is we have 124 is equal to 100 plus 20 plus 4. Now these numbers 120 are divisible by 2 and this number is 4. So we say that 124 is divisible by 2. Next we have divisibility by 9 and 3. In all these 3 cases that is in the divisibility of the numbers by 2, 5 and 10 we see that the divisibility is decided by the ones digit. But for checking the divisibility of a number by 9 this will not work. We say that a number N is divisible by 9 the sum of its digits divisible by 9. Otherwise we say that it is not divisible by 9. Also we have that a number N is divisible by 3 if the sum of its digits is divisible by 3 otherwise it is not divisible by 3. Like if you consider a number C B A this is written as 100 C plus 10 B plus A and we can rewrite this as 99 C plus 9 B plus A plus B plus C. And this is equal to 9 into 11 C plus B plus A plus B plus C. And this number 9 into 11 C plus B is divisible by 3 and 9. Hence we have that divisibility by 9 or 3 is possible if A plus B plus C is divisible by 9 or 3. Let's consider the number 927. Let's see this number is divisible by 9 or not. First let's find out the sum of its digits that is we have 9 plus 2 plus 7 is equal to 18. Now this 18 is obviously divisible by 9 that is we have that the sum of the digits of the given number are divisible by 9. So we say that 927 is divisible by 9. In the same way we can find out whether a given number is divisible by 3 or not. So this completes the session. Hope you have understood the tests of divisibility.