Textbook

Imagine a square chessboard with checkered dark and light squares adjacent to each other. Now let’s say you place a piece on a dark square. Let us call that the 5th square on a specific column. If you were to move one square up or down the column, you would naturally be putting the piece on an adjacent, light square. If you were to move two squares up or down the column, you would put the piece on a similar, dark square. In essence, if you are to move an even number of squares, you end up on a similar, dark square, and if you move an odd number of squares, you end up on an opposite, light square. This is an automatic and natural way to determine if a square is dark or light, even if it is quite far from the original square.

Similarly, this principle can be used with odd and even numbers. You’ll save time on quant problems if you just memorize these simple odds and evens rules.

**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 $×$ odd = odd
- Even $×$ even = even
- Odd $×$ even = even
- Even $×$ odd = even

**Division:**

- Even or odd depending on the values

Division is a special case because the values could have any number of factors. Consider $2/2=1$, but $4/2=2∗2/2=2$.

**Exponents**

- Expand the exponent, then use multiplication/division rules

We’ll cover rules around how to manipulate exponents in a later chapter. For now, just remember that an exponented number can be expressed as simple multiplication or division and solved one step at a time. For instance:

$x =(−2)_{3}=−2∗−2∗−2=−8 $

If it seems daunting to memorize all these rules (admittedly, they all look very similar), you can simply use $1$ and $2$ as a basepoint to derive each of these rules. For example, if you need to know what (odd) $×$ (even) equals, simply ask yourself: *is $1×2$ odd or even*? Since $1×2=2$, the answer would be even. These rules are consistent no matter how large the number is, so you can feel confident deriving these rules even with very simple numbers.

These rules can be used to solve certain kinds of quant problems quickly. For example:

$p=(124+352)×(252+139)$

Is the value for $p$ odd or even?

Without putting any of the above into a calculator, you could easily solve if $p$ is odd or even. What do you think? If you determined that the left parenthesis must be even and the right parenthesis must be odd, then you are on the right track! With that information, we can determine that this is an even number multiplied by an odd, which is, of course, even.

A trickier version of this question uses variables instead of actual numbers.

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

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

- $2a−b$
- $ab−a$
- $b_{2}−a_{2}$
- $2ab +a$
- $(b+a)_{2}$

Give it some thought, and then check your answers.

(spoiler)

Solutions

- Even
- Odd
- Odd
- Varies (try $a=3,b=2$ or $a=3,b=4$)
- Odd

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

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