   SEARCH HOME Math Central Quandaries & Queries  Question from Megan, a student: I need to write a mathematical explanation of why this works! Start with a four digit number. (a positive integer, and all digits can NOT be the same. At least one must be different) Rearrange that four digit number. Subtract the smaller 4-digit number from the larger. 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. Now write down the next multiple of 9 that is larger than the sum. Subtract the Sum from the multiple. (multiple - sum of digits) Report Difference = to number circled. The resulting number should be the number that originally circled. Why does this work, why is the number of digits 4 rather than some other number, and why can't 0 be circled? Thanks, Megan 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.

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     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.