The Printing Press Number
by Martin Gardner
Write the number 12345679 on the board (note the missing 8).
Ask the student to name any digit from 1 to 9. Suppose it is three.
Tell the class that if they multiply 12345679 by 27, they will get a big surprise.
Children love surprises. You can see the astonishment on their faces when they get
the answer 333333333.
Repeat the stunt with a different digit, say 8.
Now the multiplier is 72 and the product is 888888888.
How do you know what multiplier to give?
Dividing 111111111 by 9 yields 12345679,
so multiplying that number by 9 gives 111111111.
If 111111111 is multiplied by any digit A, the product will be AAAAAAAAA.
So, the multiplier is 9A, where A is the chosen digit.
Since 9 x 12345679 = 111111111,
Then A x (9 x 12345679) = AAAAAAAAA or
(9A) x 12345679 = AAAAAAAAA