Every composite number is the product of some factors and also the sum of the same numbers.

On the sound of it, the assertion may appear strange. Every one knows that 4 = 2×2 and also 2 + 2 = 4. If one of the factors grows, then the product grows faster than the sum, making one of the equalities impossible: for x > 2,

Taking more factors does not help ... Or, may be it does. It's not stipulated by the problem that the factors ought to be greater than 1. The unit factors, of course, do not affect the product. They do affect the sum and, taken in a suitable number, may make up for the difference between the product and the sum of the larger factors.

For example, 36 = 9 × 4, but 9 + 4 = 13, only. Which is 23 short of the product. We solve the problem by adding 23 unit factors:

and getting the required sum:

Another example: 1001 = 7 × 11 × 13. Here we'll need 1001 - (7 + 11 + 13) = 970 units:

Copyright © 1996-2008 Alexander Bogomolny