Hi Megan,

Many of these math tricks depend on casting out nines.

Take any positive integer n, divide it by nine and record the remainder. Call it r1.

Take the same integer n and add its digits. Divide the sum of the digits by 9 and record the remainder. Call it r2.

The conclusion is that r1 = r2.

There is an explanation of why this works in our response to a question from Kelera. Kelera asked about divisibility by 9 and we used the example 5634 where r1 = r2 but the explanation is the same for any remainder.

Here is your procedure.

Start with a four digit number.
Rearrange that four digit number.
Subtract the smaller 4-digit number from the larger.

Since the first two numbers have the same remainder when divided by 9 (they have the same digits) their difference will have a remainder of zero when divided by 9. So the number you have at this point is a multiple of 9.

Now circle one digit. (canNOT be zero, because that is already a circle)
Now re-write that number excluding the circled digit.
Compute the sum of the digits. (call this sum s.)

Suppose the digit you circle is d then s is d less than a multiple of 9.

Now write down the next multiple of 9 that is larger than the sum.

This is then s + d.

Subtract the Sum from the multiple. (s + d - s)
Report Difference = to number circled.

If you go to the Quandaries and Queries page and use the Quick Search for the term casting out 9 you will find some other math tricks that rely on casting out nines.

Penny