Odds, Evens and Signs: The Rules That Do the Math FOR You
Here's a scene I watch play out every week. Say you hit a Quant question that reads "If x is odd and y is an integer, which of the following must be even?" You grab the pad, write x = 2k + 1, and start pushing algebra around. Two and a half minutes later you're still expanding. Meanwhile, someone who knows the rules plugs in x = 1 and y = 1, then x = 1 and y = 2, and is DONE in about 40 seconds. Same answer. Very different amount of time left for the rest of the section.
Odds, evens and signs are some of the cheapest points on the GMAT. The rules are short. They never change. And once you know them COLD, they do the math FOR you.
Why this matters more than you think
The Quant section gives you 31 questions in 62 minutes, split between Problem Solving (regular multiple choice) and Data Sufficiency (more on that below). That's an average of 2 minutes per question. The test is adaptive question by question, so once you answer, you can't go back and fix it.
Now think about what that means. Every minute you burn doing algebra on a question that a rule could have answered is a minute you steal from a harder question later. Quant is not a math test. It's a thinking test that uses math as the gauge. These questions are checking whether you can REASON about numbers, not whether you can grind through them.
The rules, all of them
First, two terms. An integer is a whole number with no fraction or decimal part: 7, 0 and negative 3 are all integers. An even integer divides by 2 with nothing left over. An odd integer doesn't. Two facts people forget: ZERO is even, and only integers can be odd or even. A number like 2.5 is neither.
| Operation | Rule |
|---|---|
| Even + Even | Even |
| Odd + Odd | Even |
| Even + Odd | Odd |
| Even × any integer | Even |
| Odd × Odd | Odd |
Subtraction follows the same rules as addition. Powers follow multiplication: an odd number raised to a positive whole number power stays odd, and an even number stays even.
Now signs. Zero is neither positive nor negative. For multiplication and division:
| Operation | Rule |
|---|---|
| Positive × Positive | Positive |
| Negative × Negative | Positive |
| Positive × Negative | Negative |
Stop reading for a second and look at that table. What's REALLY going on? Here it is: count the negatives. An even count of negative numbers multiplied together gives a positive. An odd count gives a negative. So the sign rules are just the odd and even rules wearing a different jacket. That's also why an even power hides the sign (b² is positive whether b is 3 or negative 3) and an odd power keeps it (b³ has the same sign as b).
That's the whole list. You could write it on an index card. But knowing the rules is only half the job. The other half is PROVING them on the question in front of you, and that's where testing values comes in.
Rules plus test values: a Problem Solving example
Notice what happened. Two rounds of tiny arithmetic, no algebra. I'll bet that's faster than the way you'd have done it a month ago.
Signs in Data Sufficiency
Quick explainer if you're new to it. A Data Sufficiency question gives you a question and two statements. You don't have to find the answer. You decide whether the information is ENOUGH to answer it. The answer choices are always the same five.
The 5 mistakes I see every week
Forgetting zero. People test 1, 2 and 3 and never think about 0, which is even and has no sign. Questions about "must be positive" love to break on zero.
INSTEAD: whenever the question allows it, make 0 one of your test values on purpose.
Applying odd and even rules to non-integers. If the question doesn't say "integer," the number could be 0.5. Odd and even don't apply to it at all.
INSTEAD: before you use a single rule, check your notes for the word "integer." No integer, no odd/even shortcut.
Testing ONE set of values and calling it done. One test tells you what CAN happen. It doesn't tell you what MUST happen.
INSTEAD: test once, then change one thing (odd to even, positive to negative) and test again. Two rounds kills most wrong answers.
Doing it in your head. "It's just odds and evens, I don't need to write it down." THAT is how you flip one sign and lose a point you'd already earned.
INSTEAD: write every test value and every result on the pad. It takes seconds and it catches the silly mistakes that actually cost you points.
Reaching for algebra first. Strong math students do this most. Writing x = 2k + 1 works, but it's the long way, and the long way is where pacing problems come from.
INSTEAD: lead with the rules and small test values. Save the algebra for when the rules genuinely run out.
Which do you want MORE: the satisfaction of a clean algebraic proof, or the 90 seconds you'll need on question 27? You only get to pick one. The Test Takers who score at the highest levels pick the time, every single time.
Your challenge for today
Grab a blank sheet. Without looking at this post, write out every odd/even rule and every sign rule from memory. Then prove each one with two sets of test values you choose yourself, including at least one zero and one negative. If you can't rebuild the full list in five minutes, you don't know it cold yet. Do it again tomorrow. Then go back to the last 10 number property questions you got wrong in practice and redo each one using ONLY rules and test values. Time yourself. I'll bet the clock tells you something your score report never did.