Imagine a square chessboard with alternating dark and light squares. Suppose you place a piece on a dark square - say, the 5th square in a particular column.

• If you move one square up or down the column, you land on an adjacent light square.
• If you move two squares up or down the column, you land on a dark square again.

So:

• Moving an even number of squares keeps you on the same color.
• Moving an odd number of squares switches you to the opposite color.

This gives you a quick way to determine whether a square is dark or light, even if it’s far from where you started.

You can use the same idea with odd and even numbers. Memorizing these few simple rules will save you time on quant problems.

Addition:

• odd + odd = even
• even + even = even
• odd + even = odd
• even + odd = odd

Subtraction (it’s basically the same as addition):

• odd - odd = even
• even - even = even
• odd - even = odd
• even - odd = odd

Multiplication:

• odd $\times$ odd = odd
• even $\times$ even = even
• odd $\times$ even = even
• even $\times$ odd = even

Division:

• Even or odd depending on the values

Division is a special case because the result depends on the factors of the numerator and denominator. For example, $2/2=1$, but $4/2=2\ast 2/2=2$.

Exponents:

• Expand the exponent, then use multiplication rules

We’ll cover rules for manipulating exponents in a later chapter. For now, remember that an exponent means repeated multiplication (and sometimes division), so you can evaluate it step by step. For instance:

$$\begin{array}{rl}x& =(-2{)}^3\\ & =-2\ast -2\ast -2\\ & =-8\end{array}$$

If memorizing all these rules feels confusing (many of them look similar), you can always re-derive them using simple numbers like $1$ and $2$. For example, to figure out (odd) $\times$ (even), ask: Is $1\times 2$ odd or even? Since $1\times 2=2$, the product is even. These rules hold no matter how large the numbers are.

These rules let you answer certain quant questions quickly. For example:

$p=(124+352)\times (252+139)$

Is the value for $p$ odd or even?

You don’t need a calculator. Determine the parity of each parenthesis:

• $124+352$ is even + even, so it’s even.
• $252+139$ is even + odd, so it’s odd.

So $p$ is (even) $\times$ (odd), which is even.

Odd and even problems with variables

A trickier version of this idea uses variables instead of specific numbers.

Which of the following expressions are odd, and which are even?

Given: $a$ is odd and $b$ is even

1. $2a-b$
2. $ab-a$
3. $b^2-a^2$
4. $\frac{ab}{2}+a$
5. $(b+a{)}^2$

Give it some thought, and then check your answers.

Common themes

• If you ever forget a common arithmetic rule about odds and evens, derive it again using $0$ and $1$. For example, an odd plus an even is odd because $1$ plus $0$ is $1$.
• All even numbers are divisible by $2$ and the only even prime number is $2$.
• Any two consecutive integers are made up of both an even and an odd number.
• All odd squares have an odd square root, and all even squares have an even square root.

Bringing it all together: question walkthrough video

Here’s a walkthrough of another Achievable odd/even problem for additional review: