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A Gentle Introduction to Casting Out 9's


A mathematics teacher asks a student of her class to choose a digit other than 0. For example, 5...

Multiply it by 19. For example, 5 x 19 = 95....

Add the digits in the product. For example, 9 + 5 = 14....

And then tell her this sum. The teacher can then instantly (i.e. magically) tell the student the chosen digit. For example, somehow she knows from 14 that it is 5!

What is her method (other than multiplying out and memorizing the nine possible products)?

Why are the digits 6 and greater no fun?

Can you prove why this process works?

 


Hint: Try all nine possibilities....

Also, note that in some cases, you may need to sum the obtained digits twice to get a single digit answer...

 


Solution Commentary: Now, for a proof...or reason that the "trick" works.

Suppose your digit was A. Then A x 19 = A(10+9) = 10A + 9A = (1+9)A + 9A = A + 18A.

Notice that 9 divides the 18A...this is a hint at the idea of the Casting out 9's technique. Study the page...until you see the connection.