You must have had a friend who must have told you to think of a number and do various things to it, and then without knowing your original number, telling you what number you ended up with. Here are two examples. Examine why they work.

(i) Choose a number. Double it. Add nine. Add your original number. Divide by three. Add four. Subtract your original number. Your result is seven.

(ii) Write down any three-digit number (for example, 425). Make a six-digit number by repeating these digits in the same order (425425). Your new number is divisible by 7, 11 and 13.

(i) Steps :

1. Let our original number be n.

2. Double the number (2n).

3. Add nine to the number (2n + 9).

4. Add original number (2n + 9 + n = 3n + 9).

5. Divide the number by 3 = $\dfrac{3n + 9}{3} = \dfrac{3(n + 3)}{3}$ = n + 3.

6. Add four (n + 3 + 4 = n + 7).

7. Subtract original number (n + 7 - n = 7).

From above steps,

It is clear that on selecting any number, the answer remain same, i.e., 7.

(ii) Multiplying 7, 11 and 13 we get :

7 x 11 x 13 = 1001

Take any three digits number say, xyz.

On multiplying,

⇒ xyz x 1001 = xyzxyz

Since, 7, 11 and 13 are factor of 1001 and 1001 is a factor of xyzxyz.

∴ 7, 11 and 13 are factor of xyzxyz.

Hence, any six digit number of form xyzxyz formed from a three digit number of form xyz is divisible by 7, 11 and 13.