Question

Sum of some positive integers is 3841. If P is the product of those numbers, what is the maximum value of P?
Briefly explain.

Answer 1

This is my own creation/idea.  I Hope this is valid universally for all numbers like 3841 above.

   Divide the number in to N  integers such that each part is nearly equal to e (Euler number 2.718..).   The product will be the maximum.   Since the closest integer number to e is 3.  Divide  3841  into 1279 parts of value 3 each and two more parts of value 2.   After 3, the nearest integer to e  is  2. 

   So  the sum is:    3 + 3 + 3 + .... 1279 times  + 2 + 2 = 3831

The product of these numbers which is maximum for any partition of 3841:
      3¹²⁷⁹ * 2 * 2  ≈  6.9 * 10⁶¹⁰

You can check up the product of any other partition.  It should be less than this.

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  I hope the following generic solution for any integer sum S is formal and is a valid proof.  I give the proof here below.

Let us divide the number (an integer sum) S as :
        x1 + x2  + x3 +  x4 + .... + xn = S      where    1<= x_i <= S

=>  xn = (S  - x1 - x2 - x3 - .... x_n-1)

Product: P  =  x1 x2 x3 x4 .... x_n-1 * (S - x1 - x2 - x3 .... - x_n-1) = S

Differentiate partially wrt x1, then partially wrt  x2,  then partially wrt x3 and so on.. Equate them to 0 for maximization.

dP/d x1 =   (x2 x3 x4 ...x_n-1) [S - 2 * x1 - x2 - x3 ....- x_n-1] = 0
             =>    2 x1 + x2 + x3 + .... + x_n-1 = S
         =>  x1  = xn

Similarly when we do  dP/d x2,  we get  x2 = xn      and  so  on...

So  the product is maximum when   all parts (partitioned parts) are equal.

Now we have to find out the number "S/n" of each part and number "n" of parts in to which S is to be divided.
 
   Sum = S =  n * S/n

   Product = P = (S/n)^n  = S^n / n^n

To find the optimum value of n,  differentiate  p wrt to n:

     Ln P = n Ln S - n Ln n
  =>   1/P * d P/ dn =   Ln S  - n  * 1/n  -  Ln n
                               =    Ln (S/n)  - 1
   Equating the derivative to 0, we get  S/n  = e  (Euler e = 2.718..)

Thus each part is equal to e and there are    S/e parts into which S is divided.

Since  this question is integer based, we will have parts equal to 3  if the number is completely divisible by 3.  Otherwise, we will have either one part or two parts of value 2.  

Answer 2

Hello')

The answer:2

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