A Puzzle of Numbers

           It was a rainy Sunday afternoon. I took shelter inside a friend’s house. He was entertaining a group of people. I joined the group. We were discussing numbers and their interesting qualities. Then my friend who is a mathematician said that he would show us something very interesting.

He gave me a piece of paper and asked me to write any three digit number.

‘Can there be any zeros in it? ‘I asked.

 ‘Any number, using any digits from zero to nine. But don’t show me the number’ he said.

I wrote down a three digit number and asked him what to do next.

‘Fold the paper and pass it on to the man next to you’ he said.

‘What do I do? ‘Asked the man next to me.

‘Write the same number alongside and pass it on to your neighbour’ he said.

‘Now you’ve got a six digit number. Divide this number by seven ‘he said to the man who had the paper.

 ‘What if it doesn’t divide? What if it leaves fraction?’ asked that man.

‘It will, don’t worry’ said my friend.

‘But how do you know? You haven’t even seen the number’.

‘Leave that to me. Just divide, tear a piece of paper write the result on it and pass it on to the man next to you.’

When the next man got the number, my friend asked him to divide the number by 11 and pass on, only the result to the next man. The next man was now asked to divide the number by 13.

‘This time, I am sure the number will not divide by 13. Very few numbers do ‘he said.

‘That’s my headache, you just go ahead and do the division’ said my friend.

‘Good god. It does divide by 13! Was just lucky remarked the man with the slip.

‘Now write down the result in another bit of paper Fold it many times over so that I do not see the number and give it to me’ said my friend.

When he got the folded bit of paper, he handed it over to me and asked, ‘It this the number you wrote down to start with?’

I was amazed! It was exactly the three digit number I had written at the outset.

How do you explain it?

Answer:    First of all, let us see what happened to the original number. A similar number was written alongside it. It worked out to the same as if we took a number, multiplied it by 100 and then added the original number. For example:

872872 = 872000 +872

Here my friend has actually multiplied the original number by 1001. What did he do after that?

He had it divided successfully by 7, 11 and 13 or by 7x 11x 13 i.e. by 1001.

So he actually first multiplied the original number by 1001 and then had it divided by 1001.

How very simple!