Lesson Product of Consecutive Positive Integers

Problem: The product of two consecutive positive integers is 650. What are the integers?


Solution: Two consecutive positive integers can be defined as 'x' and 'y', but then you have two unknowns. That means you will need two equations to find a solution. However, given the two integers are consecutive, we know that y = x+1. So, the better way to define to consecutive integers is as 'x' and 'x+1'. Then the solution will go smoothly.

The product is defined by: x*(x+1) = 650

Solving the resulting quadratic will determine if there are two consecutive positive integers or not.

x^2 +x -650 = 0

To factor, we need to find two multiplicands that a 1 apart and total 650. A good place to start is the square root of 650. Using a calculator, we find that:

sqrt(650) = 25.4950976

Testing 25 and 26, we find their product is 650. (Of course, we notice that this must be our answer, but we can continue the formal solution.)

(x+26)(x-25) = 0

That means x = -26 or +25.

The problem calls for consecutive positive integers, so x=25 is chosen.

x+1 = 25+1 = 26


Answer: The two consecutive positive integers are 25 and 26.


Postscript: Of course, if we were looking for consecutive integers, then the answer would include -25 and -26.