Math Is Fun Forum

Toast

Real Member

Registered: 2006-10-08

Posts: 1,321

Odd Problem

We got this from our P.E teacher when we had nothing to do in class, so here goes:

Pick a 4 digit number where the same digit is not repeated:
e.g

1234

Now find the number which is made up of the digits reversed:

4321

Now subtract your original number from it:

3087

Multiply the result by any number from 1 to 9, e.g:

3087*4
= 12348

Now circle any digit in the number and say the other digits of the number. Given the other digits, another person would be able to solve for the circled digit. How is this so?

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Odd Problem

Your original 4-digit number can be written as 1000a + 100b + 10c + d.

Reversing the digits changes this into 1000d + 100c + 10b + a.

Subtracting one from the other gets you 999d + 90c - 90b - 999a, with the minus sign removed if there is one.

As 999 and 90 are both multiples of 9, it follows that the difference between your two numbers is also a multiple of 9.

Multiplying the difference by another number will not affect this.

A property of the multiples of 9 is that their digits all add up to 9. So your teacher would know that whatever your number is will always have digits that total a multiple of 9.

As you tell your teacher all the digits except one, they could add those digits up and work out what the other digit must be for the total to be a multiple of 9.

With your example of 12348, you could, for example, circle the number 4 and tell your teacher the numbers 1, 2, 3 and 8.

The teacher would then add these up to get 14, and work out that 14 needs 4 more to become 18, a multiple of 9. And so they would know that the missing number is 4.

Incidentally, next time your teacher tries that on you, try to get a result with a 0 or a 9 in it. If you pick that one and tell your teacher the others, they won't be able to give you a certain answer!

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Why did the vector cross the road?
It wanted to be normal.