Once in a while, the newspaper talks about youngsters or adults who have the seemingly incredible ability to perform arithmetic computations in their head for large integers. Actually, this all boils down to a basic understanding of algebra and algorithmic thinking. Yes, we can all do it. Just to give you an example, suppose we wish to know 35×35. An algorithm for getting the right answer says:
“the last two digits are 25; and the digits in front are equal to the product of the first digit with itself added 1”
This gives the answer 1225, since the first digit is 3, and 3×4 = 12. To see why this algorithm is true, we first look at the simple case where there is only one digit in front of 5. The product of two integers with 5 as the last digit and a as the first digit can be written as:
a5 x a5 = (a x 10 + 5)(a x 10 + 5) = 100a^2 + 50a + 50a + 25 = 100a(a+1) + 25
The first term has two zero digits at the end, so that when 25 is added to it, it is clear that the last two digits are just 25. Finally, the digits in front are just a(a+1).
Now you should be able to figure out an algorithm for multiplying something like 85 x 35, and perhaps, later, more general things like 29 x 29, and finally 73 x 48. Have fun!