Number Theory Refresher for SAT Math

The term “number theory,” sounds like some abstract concept – the kind of thing you’d study in an advanced math class in college. However, number theory is a foundational part of mathematics, and whether you realize it or not, you already know many of its basic tenets.

Number theory is the study of natural numbers and integers. Natural numbers are numbers you use to count with (for example there are 365 days in one year) and integers are positive or negative whole numbers. For example, 3, 44, -7, and 5 are all integers while 9.64, 25/3, and square root of 2 are not.

The SAT expects you to know many of the rules and definitions of number theory. Many of these rules and definitions will be familiar, but it’s best to review them to make sure you know them by heart.

Let’s start with a few common terms and their definitions.

Integer

A positive or negative whole number, for example 5, -2, 47.

Factor

Factors are numbers you multiple together to get another number. For example, the number 12 has the factors 3 and 4.

3 x 4 = 12

Multiple

A multiple is the product of any quantity and an integer. For example, 10 is a multiple of 2 because 2 times 5 (another integer) equals 10.

Prime number

A prime number is any number that only has two factors: 1 and the number itself. For example, 7 is a prime number: no two numbers other than 1 and 7 can be multiplied to equal 7.

Other prime numbers include 2, 3, 5, 11, 13, 17, 19, 23, 29, and 31.

Absolute value

Absolute value is best visualized as the distance between a number and zero – in other words, how many numbers are between a certain number and zero. For example, the absolute value of 5 is 5 and the absolute value of -5 is 5 because both 5 and -5 are the same distance away from zero. If you find this confusing, this visual will help you understand.

 

Simple number theory rules

In addition to these definitions, which you’ll see frequently on math questions, you should also commit the following rules to memory.

Odd and even numbers

An even number is a number that is divisible by two. In other words, every even number is exactly twice another (even or odd) integer. An odd number is a number that can’t even. Ok, not really. An odd number is simply an integer that does not divide evenly in two. Given these definitions, there are predictable patterns on the SAT that you should know. See below:   

Odd

±

Odd

=

Even

Even

±

Odd

=

Odd

Even

±

Even

=

Even

Odd

×

Odd

=

Odd

Even

×

Odd

=

Even

Even

×

Even

=

Even

 

Positive and negative numbers and absolute value

SAT questions will often test your ability to work with positive and negative numbers. Positive numbers are greater than zero and negative numbers are less than zero. Remember, zero is neither negative nor positive itself. Here are additional  rules you should know:

Adding a negative number is the same as subtracting a positive number. For example:

4 + (-5) = 4 – 5

When multiplying or dividing two numbers of the same sign (both negative or both positive), the results will always be positive. On the other hand, the product or quotient of a mixed pair will always be negative.

(-2)(-2) = 4

(-2)(-2)(-2) = (4)(-2) = -8

Remember the definition of absolute value above? Though absolute value is pretty simple, it can easily trick you when applied to variables. Algebraic expressions inside absolute values will often have two solutions. For example, |-6| = 6, but the number with an absolute value of |y| could be y or –y.

Not all of the above definitions or rules will be the subject of an SAT math question, but on many questions, you’ll need to know these rules and definitions in order to solve for the answer. Be sure to review these rules and know them like the back of your hand