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All whole numbers (including 0) and their negative counterparts
Integers are infinite (do not include infinity specifically)

How we plot/visualize numbers.
Increases moving left to right.
Includes all real numbers.
Includes rational and irrational numbers.

The base number times itself the value of the exponent.
A shorthand form of multiplication.

A number, when multiplied by itself by the # of times of the root, produces the number "under the house".
It is essentially a "canceling out" or reversal of exponents.

Shorthand multiplication of a consecutive sequence of numbers down to 1, beginning with the X!

1. Negative + Negative = Negative
2. Negative + Positive = Depends on greater value

1. Negative - Negative = Depends on greater value
2. Positive - Negative = Positive
3. Negative - Positive = Negative

Negative * $\div$ Negative = Positive (cancels out)
Positive * $\div$ Negative = Negative
Negative * $\div$ Positive = Negative
For $-\frac{x}{y}$, negative symbol applies to either numerator or denominator, not both. Does not cancel out unless both have negative symbol.

$(-3)^2$ is (-3) * (-3) = 9
$-(3)^2$ is (-3) * (3) = -9 OR $-1 *(3)^2 $
$-3^2$ is (-3) * (3) = -9 OR $-1 *(3)^2 $

$(neg.)^{even}$ = positive
$(neg.)^{odd}$ = negative
$neg.^{anything}$ = negative

**There is a small, invisible 1 next to the negative symbol with no parenthesees.**
You MUST have (parenthesees) around the ENTIRE exponent for negative cancellation rules with an even exponent!

Order of operations:
Parenthesees
Factorials
Exponents
Multiplication
Division
Addition
Subtraction
[MD] & [AS] in order from L to R.

Inclusive: (Last) - (First) +1
Exclusive: (Second to last) - (Second) + 1

1. If starting from 1:
$\frac{x(x+1)}{2}$, with x being the last number in the interval

2. If starting from any other number:

Median: $\frac{last+first}{2}$ * # of Integers: last-first +1

A factor is an number that is can evenly divide another number without a remainder.
Ex. 6 is a factor of 12 because 12/6 = 2

Listing out the integers and making the gap between the two multiplied numbers smaller and smaller until you find all of them.

1. Break down the number into its prime factorization.
2. Add 1 to each of the exponents ex.$(x^{e+1})$
3. Multiply the modified exponents together (do not include the base number!)

1. Breakdown the factorial, use prime factorization and "shop for factors" to build numbers from factorial components

Any prime number greater than the factorial (X!) and its multiples.

1: Any number is divisible by 1 (except 0)
2: Any number that ends in an even unit digit
3: The sum of all digits must be divisible by 3
4: The number of the last two digits must be divisible by 4, OR are 00 (i.e 264, 300)

Even: divisible by 2, includes 0
Odd: not divisible by 2

Add/Sub:
$Even ± Even$ = Even
$Odd ± Even$ = Odd
$Odd ± Odd$ = Even

Multiplication:
$Even * Even$ = Even
$Odd * Even$ = Even
$Odd * Odd$ = Odd

Division:
$\frac{Even}{Even}$ = Both or decimal
$\frac{Odd}{Even}$ = Decimal
$\frac{Odd}{Odd}$ = Odd or decimal

Exponents:
$Even^{integer}$ = Even
$Odd^{integer}$ = Odd
$Exception: All^0$ = 1

5: Any number that ends in 0 or 5
6: Any number divisible by 2 and 3
7: Use a calculator
8: If the last 3 digits are divisible by 8, or 000
9: If the sum of all digits are divisible by 9
10: Any number that ends in 0
11: Swap signs (start positive) and addd, solution must be 0 or divisible by 11. (May be easier to just use calc here)

1. Find the square root of the number
2. Divide by all primes smaller than that number (including decimal, if its 11.1, divide by 11)
3. If indivisible by any primes smaller than the square root, it is prime.

The largest common factor between 2 or more numbers.
GCF of two primes is 1.

The products when a number is multiplied.
0 is a multiple of all integers.
LCM - smallest multiple in common between two or more integers.
LCM of two primes is their product.

If you add two multiples of a number together, the sum will also be a multiple of that number.
Ex. 30! + 29
30! is a multiple of 29/29 is a factor of 30!
29 is a multiple/factor of 29.
Therefore, the sum of 30! and 29 will also be a factor of 29.

$\frac{last-first}{multiple} +1$

$\frac{number of multiples}{2} $ * sum of one pair (of multiples)

Three integers in a row.
Can be written as x, x+1, x+2 or x-1, x, x+1

The result is ALWAYS a multiple of 3 regardless of what three consecutive integers you choose.
Odd Even Odd = Even
Even Odd Even = Odd
Rules apply for ALL consecutive integers of three, including odd/even and multiples consecutive integers

Product ALWAYS a multiple of 2, 3 and 6, regardless of what three consecutive integers you choose.
Product is always even
Product SOMETIMES a multiple of 4 and 8 (if the odd number is in the middle)
One number in the sequence always a multiple of 3
Rules apply for ALL consecutive integers of three, including odd/even and multiples consecutive integers

The denominator/the number you divide by MUST be or be converted to a prime number.
1. Prime factor the number being divided (as needed).
2. Find the number of multiples by listing $X^1$ and then divide by that number until you reach 0. If broken into two primes (ex. 6 - 2, 3), find the multiples for the limited factor, or the one with the least number of multiples. Sum together.
3. Take the sum and divide by groups (aka the exponent of the prime if you had to break down into one prime number ex. 9 to $3^2$)

Find the # of squares of 10 in the factorial.
Break 10 into prime factors (5 and 2)
Find the # of multiples for the limited factor (5).

- It is neither positive nor negative.
It is even (because it's divisible by 2)
It is a multiple of EVERY integer.
It's a factor only of itself.
x/0 = undefined
$x^0$= 1
$0^0$= undefined
0! = 1

Primes are the "building blocks" of numbers. Every positive integer greater than 1.
And of course primes only have two positive factors: 1 and itself.

There are 15 primes between 1 and 50.

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47

Every positive integer greater than 1 can be written as the product of primes in exactly one way. Thus, the Prime Factorization of an integer greater than 1 serves as its DNA, its unique identifier.

$24 = 2^33^1$

There are infinitely many primes, though the number of primes within a fixed interval tend to decrease as the numbers get larger. For example, you can see how the number of primes decreases in each interval.
Notice that, by adding 1, we created a new number that cannot share any of the prime factors of the number
1 below (30). And in so doing we found a new prime: 31

To find GCF using PF:
1. Break down each number into its prime factors
2. Identify the prime divisors the two original numbers have in common. Make sure to note the exponents they have in common too.
3. Multiply together the shared numbers to find GCF

ODD: Find total number of factors (PF, exponents +1, multiply exponents together without their bases), except ignore the even prime numbers (i.e. the bases, like 2, doesn't matter what the exponent is).
EVEN: Find total positive factors, then total odd factors, and subtract

1. Prime factor both numbers
2. Identify the numbers both have in common (inc. their exponents)
3. Do the "trick" where you add one to each exponent and multiply the numbers together.

1. Prime factor both numbers
2. Across both numbers, write down all divisors (big #'s) present
3. Find the largest exponent for each divisor. Choose the largest exponents to bases and multiply.

Common GRE problem: Three or four numbers that occur at regular intervals. All instances will occur together at some point.
"How much time will elapse before they all occur together again?"
To Solve:Find the LCM of all numbers

- 2,3,7,8: cycle of 4
- 4, 9: cycle of 2
- 0,1,5,6: cycle of 1 (always ends with their own digit)

The digit in the ones place, before the decimal. Must be an integer.

In GRE problems, remainders must always be positive. If your calculated quotient makes the remainder negative, you must adjust the quotient down to the next integer to yield a positive remainder (e.g., -17 ÷ 7 results in a quotient of -3 (-21 on the number line, and a remainder (back to -17) of 4)

What is "left over" in a division problem. Remainders can never be negative. They're either zero or positive integers.

Whenever the number being divided is less than the divisor (ex. 5/6), and both integers are positive, the remainder is simply the number being divided.

Ex. 5÷6
In this case, the remainder is 5. This is because... 0 (quotient) × 6 + 5 (remainder) = 5
How many times does 6 go into 5? Exactly 0 times, because 6 is larger than 5.

For example, what if we're doing this: − 29 ÷ 7

Mistake: − 4 (quotient) × 7 = -28 R -1 (NO NEGATIVE REMAINDERS)
Correct: − 5 (quotient) × 7 = -35 + 6 (remainder)
The reason why the first way is wrong is because it results in a negative remainder. Remember that's not allowed.

Dividing with large exponents:
HOW TO SOLVE:
$\div1$ - remainder is always 0
$\div2$ - remainder is 1 if UD is odd, 0 if it's even
$\div3$ - find UD location in cyclicity pattern, divide that by 3 to get remainder
$\div4$ - find last two digits, then use divisibility rules of 4 to get remainder
$\div5$ - calculate UD of one's base. If UD is less than 5 (0-4), that is the remainder. If UD is 5 or greater, subtract 5
$\div6$ - find UD location in cyclicity pattern, divide that by 6 to get remainder
$\div7$ - find UD location in cyclicity pattern, divide that by 7 to get remainder
$\div8$ - calculate last 3 digits being divided and use divisibility rules of 8 to find remainder
$\div9$ - find UD location in cyclicity pattern, divide that by 9 to get remainder
$\div10$ - UD is the remainder

To find the overall remainder in a Remainders and Addition problem, find the remainder of each term separately and then add them together.
If the remainder is larger than what we are dividing by (ex. 13/10), then divide the remainder sum by the divisor and get the remainder

Numbers that are written as $\frac{integer}{integer}$.
Denominator can never be 0
Factorials can be written as fractions (bc they are integers).

Rational numbers are numbers that can be written as a fraction.
Irrational numbers are numbers that cannot be written as a fraction.