For integers, the following are true where x, y, and z are any integers (actually of real numbers as well)

$x+y=y+x$ or $x*y=y*x$**Commutative Property**

$x*(y+z)=x*y+x*z$**Distributive Property**

The**absolute value** of a number x is denoted by |x| or abs(x) and is equal to the number without its sign

e.g. |-¹|=|¹|=¹. |-3.14|=|3.14|=3.14.

The**factorial** of a number $x$ is denoted by $x!$ and is equal to the product $x*(x-1)*(x-2)...*2*1$.

Also, $0!=1!=1$ by definition. So, for example, $3!=3*2*1=6$, $(5!)/(3!)=5*4=20$.

An integer*x* is said to be **divisible** by another integer *$y$* if *$x/y$* has an integer quotient and leaves a zero remainder

The integer y is called a**factor** of x, and x is called a **multiple** of y.

Integers divisible by 2 are called**even** integers. Those that are not are called **odd** integers.

Integers divisible only by 1 or by themselves are called**Prime** numbers.

**2 is the only even prime.** Non-prime integers are called **Composite** numbers.

**1 is neither prime, nor composite**.

e.g. $2, 3, 5, 7, 11, 13, 17, 19, 23$, ,,, are all prime numbers.

**Prime Factorization** of an integer is the process of expressing every integer as the product of its component primes.

Each component prime is called a**prime factor**. Some prime factors may repeat. Every number can be expressed as the product of its prime factors.

e.g. $124=2*2*31$, note that 2 is a repeating prime factor, and 2 and 31 are both prime.

Two numbers are called**relatively prime** if the only common prime factor between them is 1.

Given two integers, their**Greatest Common Divisor (GCD)** is defined to be the product of their common prime factors.

e.g. $36=2*2*3*3$, $132=2*2*3*11$, so the set of common factors to 36 and 132 is {2, 3, 4} so the GCD of 36 and 132 is $2*3*4=24$

From the foregoing we see that relatively prime numbers have a GCD of 1.

**Rules for divisibility**

Might seem daunting at first, but most of these are common sense.

As is typical, we only look at divisibility rules for common primes, because we usually apply these rules to find prime factors.

A number is divisible by 2 if it is even.

A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisible by 5 if the last digit is a 5 or a 0.

A number is divisible by 11 if the difference of the sums of alternate sets of digits in the number is divisible by 11

A**multiple** of an integer is created by multiplying that integer by another integer.

e.g. $3, 6 (=3*2), 9 (=3*3)$, ... are multiples of $3$. $2, 4 (=2*2), 6 (=2*3)$, ... are multiples of $2$.

Notice how $6$ is a multiple of both $2$ and $3$. This is called a**Common Multiple** of 2 and 3.

The smallest such common multiple of two numbers is called the**Least Common Multiple (LCM)**

e.g. The LCM of 2 and 3 is 6 (see above). That of 4 and 8 is 8. (for 4 we have 4, 8, 12, 16, ...; for 8 we get 8, 16, 24, ...).

**Tricks with GCD and LCM calculations**

If GCD(x,y) denotes the GCD of x and y, and LCM(x,y) denotes their LCM, then**GCD(x,y)*LCM(x,y)=$x*y$**

**Euclid's Algorithm**: If $x=q*y+r$, where $q, r, x, y$ are all integers, then **$\text"GCD(x,y)=GCD(y,r)"$**

e.g. $\text"GCD(119,49)=?"$ If we know $119=7*17$, then we can say GCD=$7$ easily.

If we don't know that though, we can still use Euclid's algorithm since $119=49*2+21$ to say $\text"GCD(119,49)=GCD(49,21)=7"$ much more easily.

$x+y=y+x$ or $x*y=y*x$

$x*(y+z)=x*y+x*z$

The

e.g. |-¹|=|¹|=¹. |-3.14|=|3.14|=3.14.

The

Also, $0!=1!=1$ by definition. So, for example, $3!=3*2*1=6$, $(5!)/(3!)=5*4=20$.

An integer

The integer y is called a

Integers divisible by 2 are called

Integers divisible only by 1 or by themselves are called

e.g. $2, 3, 5, 7, 11, 13, 17, 19, 23$, ,,, are all prime numbers.

Each component prime is called a

e.g. $124=2*2*31$, note that 2 is a repeating prime factor, and 2 and 31 are both prime.

Two numbers are called

Given two integers, their

e.g. $36=2*2*3*3$, $132=2*2*3*11$, so the set of common factors to 36 and 132 is {2, 3, 4} so the GCD of 36 and 132 is $2*3*4=24$

From the foregoing we see that relatively prime numbers have a GCD of 1.

Might seem daunting at first, but most of these are common sense.

As is typical, we only look at divisibility rules for common primes, because we usually apply these rules to find prime factors.

A number is divisible by 2 if it is even.

A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisible by 5 if the last digit is a 5 or a 0.

A number is divisible by 11 if the difference of the sums of alternate sets of digits in the number is divisible by 11

A

e.g. $3, 6 (=3*2), 9 (=3*3)$, ... are multiples of $3$. $2, 4 (=2*2), 6 (=2*3)$, ... are multiples of $2$.

Notice how $6$ is a multiple of both $2$ and $3$. This is called a

The smallest such common multiple of two numbers is called the

e.g. The LCM of 2 and 3 is 6 (see above). That of 4 and 8 is 8. (for 4 we have 4, 8, 12, 16, ...; for 8 we get 8, 16, 24, ...).

If GCD(x,y) denotes the GCD of x and y, and LCM(x,y) denotes their LCM, then

e.g. $\text"GCD(119,49)=?"$ If we know $119=7*17$, then we can say GCD=$7$ easily.

If we don't know that though, we can still use Euclid's algorithm since $119=49*2+21$ to say $\text"GCD(119,49)=GCD(49,21)=7"$ much more easily.

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