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 a few simple rules will save you time on quant problems.

Addition:

• Odd + odd = even
• Even + even = even
• Odd + even = odd
• Even + odd = odd

Subtraction is 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 in the numerator and denominator. For example, $2/2=1$, but $4/2=2\ast 2/2=2$.

Exponents

• Expand the exponent, then use multiplication/division 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: