# Integers

Pythagoras, best known to high school students for his Pythagorean Theorem, actually discovered much more than that one formula. Even if you are not mathematically inclined, the beginning of this paper has some interesting notes on how the Pythagoreans--the followers of Pythagoras--lived.

## 3/28 Integers

Always attempt to solve the problem before looking at the answer choices. Read carefully, then identify the bottom line--what the question is actually asking--and mark it at the top of your scratch work. Assess you options by asking "What could I do?" to open your toolbox, then "What should I do?" to select the best way to solve the problem. Attack the problem fearlessly, then loop back to the bottom line to check whether what you found is the correct answer.

If p is an odd integer, which of the following is an even integer?

At the top of your scratch work, write even = ?

Next, ask "What could I do?" You could think through each answer choice abstractly, determining that if p is odd then... but that is difficult and gets confusing quickly. You could pick a number for p, then use that number to find a value for each answer choice. The smallest odd number is the best for this. Pick one. Since this question includes the phrase "which of the following," the answer is very likely to be D or E. Start at the bottom and work your way up.

E) $p^{2}-p$
If p = 1, then $1^{2}-1=0$. 0 is neither positive nor negative, but neutral; however, it is still even. This distinction confuses some students, so make sure you know it. Now loop back to the bottom line. $p^{2}-p=0$, so it is even, so it is the answer. On the SAT, you could continue on from this point or check the other answers.

D) $(p-2)^{2}$
$(1-2)^{2}=(-1)^{2}=1$ is odd.

C) $p^{2}-2$
$1^{2}-2=1-2=-1$ is odd.

B) $p^{2}$
$1^{2}=1$ is odd.

A) $p-2$
$1-2=-1$ is odd.