What is the maximum product of two numbers whose addition is 17 where the two numbers are integers?

Question

It is very obvious that the numbers are 8.5 and 8.5 if it is not integer
x+y=17 y=17-x
Let t=xy=x(17-x)
Max of t is 72.25
If it is integer the answer is 9 and 8 and the maximum product is 72
My question is to know is there any mathematical method to solve if the numbers are integers.

Answer 1

We can use the inequality of arithmetic and geometric means.

$$\frac{x+y}{2} \geqslant (x\cdot y)^\frac12$$

Plugging in $x + y = 17$, and squaring both sides of the inequality, we get,

$$(\frac{17}{2})^2 \geqslant (x\cdot y)$$

which is reduced to

$$72.25 \geqslant x\cdot y$$

Now you can find the greatest integer less than or equal to the quantity on the left hand side of the above inequality and find the factors (that multiply to give this integer) which have the least difference in magnitude.

In this case, for instance, we get 8*9 = 72

In case the greatest integer less than or equal to the quantity on the left hand side is a prime number, then find the next integer less than it and repeat the above step.

EDIT: This is only an elementary way to find the answer; it also fails when the sum is negative.

Please look at the other answers, they're shorter and more efficient.

Answer 2

We want to maximize $$x(17-x)$$ which is a concave quadratic function and it is symmetrical about $x=8.5$ and maximal at that value without constraint.

The closer we are to $8.5$, the higher is our value, hence for integer value, we have $x=8$ or $x=9$.

Answer 3

Without restriction of them needing to be integers the parts are $8.5,8.5$ as each is a half of $17$.

But when we need to choose integers we have to use the method of rounding off to nearest integer keeping their sum constant, which would now be $ 8,9$. There is no need to a priori seek a condition that should be obeyed before partitioning itself.