Here's a good example of a problem involving integers as might be seen on an SAT.

If n is an integer and n² is a positive integer, which must also be positive?

a. n² + n

b. n² – m³

c. 2n² – n

d. n³ + n

e. 2n³ + n

 

At first glance a,b and c all seem reasonable because squaring any number will give a positive answer. But answer b wont always work because if n is positive, n³ will be greater than or equal to n². So when subtracting it from n² you will either get 0, if n is 1 or negative if n >1.

For answer a, if n = -1, then n² + n = 0, which does not give a positive answer either.

Therefore answer c is the correct answer.

Any positive number squared times 2 will be greater than the number itself so 2n² – n > 0 for any positive number n.

If n is negative, 2n² is positive and subtracting a negative is like adding a positive, so 2n² – n > 0 for any negative number n as well.