Question

Write an odd natural number as a sum of two integers m1 and m2 in a way that m1m2 is maximum?

Answer 1

One integer just less than half the number and other integer just more than half the number. If the number is `2n+1`, the numbers are `n` and `n+1`.

Explanation:

Let the odd number be `2n+1`

and let us divide it in two numbers `x` and `2n+1-x`

then their product is `2nx+x-x^2`

The product will be maximum if `(dy)/(dx)=0`, where

`y=f(x)=2nx+x-x^2`

and hence foe maxima `(dy)/(dx)=2n+1-2x=0`

or `x=(2n+1)/2=n+1/2`

but as `2n+1` is odd, `x` is a fraction

But as `x` has to be an integer, we can have the integers as `n` and `n+1` i.e. one integer just less than half the number and other integer just more than half the number. If the number is `2n+1`, the numbers are `n` and `n+1`.

For example, if number is `37`, the two numbers `m_1` and `m_2` would be `18` and `19` and their product `342` would be the maximum one can have if `37` is split in two integers.