The sum of the product and the sum of two positive integers is 39. Find the largest possible value of the product of their sum and their product.

 Jan 4, 2023

Try this:


[19 x 1] + 19 + 1 ==39

 Jan 4, 2023

Sorry guest(s), I believe those are both the wrong answers. Thanks for your help, but can someone (preferably a user) please help still?

Keihaku  Jan 4, 2023

No problem, Guess I'll help...


We let one of our integers be x

and the other integer be y


xy + x + y = 39


We want to maximize the distance between the product of the sum and the product...

(Didn't really understand this part, so my answer could be wrong)

So basically, we want our sum to be big, and our product to be big, to get the largest product????


Anyway, to get the largest product, there is a trick...

Try this out yourself:

93 * 95

94 * 94


When evaluating both of them, you will notice that the second expression is greater. This is because the closer the numbers are, the greater the product.


So, when using this theory in our problem, we want the sum and the product of the two integers to be big.

In doing so, we have to think, and for this, I'm pretty sure to just use trial and error.


We have 5 and 4, product 20, sum 9, so total product 180. Pretty good... Except, that they don't sum to 39

We have 19 and 1, product 19, sum 20, so total product = 380. Pretty sure this answer is correct, from what I understand, because the numbers:

19 and 20

Are basically the closest you can go...


So, my answer is 380 as the total product.

 Jan 5, 2023

That's correct, thank you!

Keihaku  Jan 6, 2023

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