Time for MATH MINUTE! (provide your favorite theme music here). 

         Get out your paper & pencil, because I have a new puzzle for you!!

PUZZLE #107 More about nines

 

       We saw some properties of nines in Puzzle 106, here are some more.*

 

A.    As in #106, take any 2 digit number ab, a and b not equal, reverse the digits, and subtract. The result is always a multiple of 9.

       Examples:     43-34 = 9,

                             83-38 = 45 = 5x9,

                             79-97 = -18 = -2x9.

       Explain why this works.

       Actually you can use any size number of digits (why?).

 

B.    Take any 2 digit number ab, add the digits together, a+b, and subtract the result from ab, ab-(a+b). The result is always a multiple of 9.

       Examples:     13-(1+3) = 13-4 = 9,

                             43-(4+3) = 43-7= 36 = 6x9.      

       Can you explain why this works?

 

C.    Given any multiple of 9, the sum of the digits of the answer will always be a multiple of 9.

       Examples:     8x9 = 72, and 7+2=9,

                             121x9 = 1089, and 1+8+9=18=2x9.

       Why does this work?

 

D.    The sum of the digits of a number is called the digital root of the number. The number can be any size. By Part C a number is a multiple of 9 if and only if the digital root is also a multiple of 9.

       Explain why this is true.

 

E.    Casting out nines: This is a check (some call it a sanity check) on different arithmetic operations. It is not used so much with hand calculators these days.

       Problem: add up the column:      3264

                                                          8415

                                                          2946

                                                          3206

 

       First compute the excess of each number (divide the digital root of each number by 9 and list the remainder – this is the casting out operation), then add up the numbers and all of these excesses:

              3264 -> (3+2+6+4) = 9+6 ->       6

              8415 ->                                      0

              2946 ->                                      3

              3206 ->                                      2

            17831 ->                                    11

 

       Now reduce the answer and the sum of the excesses by 9:

       17831 -> (1+7+8+3+1) = 20 -> 2, and 11 -> 2.

 

       This tells us that the answer 17831 may be correct, in other words in order for the summation to be correct the two reduced sums must be equal. If the answers do not agree we know the answer is wrong!

             

       The technique also works for subtraction, multiplication, and division problems.**

       Explain why it works?

 

* See Wikipedia for more information on this:

   http://en.wikipedia.org/wiki/9_(number)

 

** http://en.wikipedia.org/wiki/Casting_out_nines

 

 

Have fun!

 

         Send your comments, ideas and solutions before Monday to the email below, and be sure to put * MM *  in the subject line:  parksjm@potsdam.edu

         Visit us here online at:  http://www2.potsdam.edu/parksjm/MM1.1.htm  to see the results every Friday/Monday.

         See you next time on MATH MINUTE!  (theme music fades out here).