Solution to "Multiplying Nines"

View Problem: Multiplying Nines

The trick lies in writing a number with K 9's (i.e. 999...9 occurring K times) as 10^K-1. So, the numbers A and B can be written as A = 10^N-1 and B = 10^M-1.

A x B = (10^N-1) x (10^M-1) = 10^N+M - 10^N - 10^M + 1

Without loss of generality, let N >= M. Let us now try to compute the above result:

   10000...0000     // N+M 0's
-     1000..000     // N 0's
-       100..00     // M 0's
+             1
-------------------------------------------------------------
[(M-1) 9's]  [1 8]  [(N-M-1) 9's]  [1 9]  [(M-1) 0's]  [1 1]
-------------------------------------------------------------

The answer can obtained by making simple observations while performing the subtractions and addition as shown above.