You have seen that a number 450 is divisible by 10. It is also divisible by 2 and 5 which are factors of 10. Similarly, a number 135 is divisible 9. It is also divisible by 3 which is a factor of 9.

Can you say that if a number is divisible by any number m, then it will also be divisible by each of the factors of m?

Write a 3-digit number abc as 100a + 10b + c

             = 99a + 11b + (a - b + c)
             = 11(9a + b) + (a - b + c)

if the number abc is divisible by 11, then what can you say about (a - b + c)?
Is it necessary that ( a + c - b) should be divisible by 11?

Write a 4- digit number abcd as 1000a  100b + 10c + d

              = (1001a + 99b + 11c) - ( a - b + c - d)
              = (1001a + 99b + 11c) - (a - b + c - d)
              = 11 (91a +9b + c) + (b + d) - (a + c)

If the number abcd is divisible by 11, then what can you say about.
[(b + d) - (a + c)]?

From (i) and (ii) above, can you say that a number will be divisible by 11 if the different between the sum of digits at its odd places and that of digits at the even places is divisible by 11?

Check the divisibility of the following numbers by 3.

1. 108 2. 616 3. 294

4. 432 5. 927

If 21y5 is a multiple of 9, where y is a digit, what is the value of y?

If 31z5 is a multiple of 9, where z is a digit, what is the value of z? You will find that there are two answers for the last problem. Why is this so?

If 24x is a multiple of 3, where x is a digit, what is the value of x?

If 31z5 is a multiple of 3, where z is a digit, what might be the values of z?

Write the following number in generalized form:

(i) 329 (ii) 502