### Instruction: What is the greatest product obtainable from two integers whose sum is 1998?### Response: I want to find two numbers that add up to 1998 and have the largest possible product.
I can use algebra to write one number in terms of the other, and then express the product as a function of one variable.
Let x be one of the numbers, then the other number is 1998 - x.
The product is then P(x) = x(1998 - x) = 1998x - x^2.
To maximize P(x), I can use calculus or complete the square to find the vertex of the parabola.
Completing the square, I get P(x) = - (x - 999)^2 + 999^2.
The vertex is at (999, 999^2), which means the maximum value of P(x) is 999^2.
This occurs when x = 999 and 1998 - x = 999, so the two numbers are both 999.