Example:
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, and so on.
Remainders
Remainders are what you get when you divide one number by another number and have something left over (this assumes you don’t use fractions or decimals to write the answer to your division problem).
Example:
If we divide 30 by 4, we see that it doesn’t work out evenly. 4 * 7 = 28, which isn’t enough, and 4 * 8 = 32, which is too much. So if we divide 30 by 4, one way to state the answer is to say that 30 divided by 4 is “7 with a remainder of 2,” because 4 * 7 = 28 and
28 + 2 = 30.
The remainder in a division problem must be less than the number we’re dividing by.
Example:
It doesn’t make any sense to say that 30 divided by 4 is “3 with a remainder of 18,” because 18 is bigger than 4 and 4 will still go into 18 a few more times.
As a reminder, when you first learned to divide, you were probably taught to use remainders.
Most calculators don’t give remainders when solving division problems—instead, they give fractions or decimals.
Prime numbers
A prime number is a number that has exactly two factors: 1 and itself.
Example:
17 is a prime number because there are no positive integers besides 1 and 17 that can be multiplied by other integers to generate 17. (Try to come up with some—you won’t be able to.)
24 is NOT a prime number because there are a lot of positive integers besides 1 and 24 that can be multiplied by other integers to generate 24. For example, 2, 3, 4, 6, 8, and 12 can all be multiplied by other integers to generate 24.
All prime numbers are positive.
The only even prime number is 2.
1 is NOT a prime number because it has only one factor (itself), while prime numbers must have exactly two factors.
Ratios, proportions, and percentages
Ratios, proportions, and percentages are all ways to express a relationship between two numbers.
A ratio is written as a pair of numbers with a colon between them.
Example:
If you make 5 dollars for every 1 dollar Bob makes, then the ratio of your pay to Bob’s pay is 5 : 1.
A proportion is usually written as a fraction, with a number in the numerator compared to the number in the denominator.
Example:
If you make 5 dollars for every 1 dollar Bob makes, then your pay can be compared to Bob’s pay with the proportion 5/1. (Or, if we wanted to compare what Bob makes to what you make, that proportion would be 1/5.)
A percentage is a special proportion where one number is compared to 100.
To determine a percentage, first compare two numbers with a proportion, and then divide the top number by the bottom number and multiply the result by 100.
Example:
If Bob makes 1 dollar for every 5 dollars you make, then the proportion that compares Bob’s pay to your pay is 1/5. If we divide 1 by 5 and multiply by 100, we see that Bob makes 20% of what you make.
Ratios can be set equal to each other and “cross-multiplied.” (If you don’t already know how to do this, don’t worry—it’s just a short cut around regular algebraic techniques. You don’t have to know how to do it for the SAT.)
If the relationship between two quantities is the kind where increasing one quantity results in an increase in the other quantity, then we say those two quantities “vary directly” or are “directly proportional.”
Example:
If I make 1 dollar for every 5 dollars you make, then when I make 4 dollars you make 20 dollars—increasing my pay to 4 leads to an increase in your pay to 20. That means our two rates of pay are in direct proportion.
If two quantities are related so that increasing one decreases the other, then we say those two quantities “vary indirectly” or are “inversely proportional.”
Example:
If we have two quantities x and y set up so that xy = 20, then x and y are inversely proportional—every time one increases, the other one decreases, and vice-versa. So if x starts out as 10 and y starts out as 2, changing x to 5 means we have to change y to 4—as one decreases, the other increases.
Sequences
Sequences are strings of numbers that follow a rule, so that knowing one number in the sequence allows us to figure out another number in the sequence.
Sequence questions on the SAT will rarely operate in exactly the same way that a question about an arithmetic or geometric series would work in a math class, though the College Board often tries to mislead you by making a sequence question look deceptively similar to a traditional question about a series.
SAT sequences can either go on forever or stop at some point, depending on the setup of the question.
There are two common types of SAT sequences, and we can classify them by the rules that are used to figure out which numbers go in the sequence. Let’s look at the different types of SAT sequences: