When I write {`A` is divisible by `B`} in this writeup, I mean {`A` modulo `B` = 0}. Essentially this means that **only integers are considered**, and if the division results in a remainder, then the division "doesn't work." It is true, of course, that, if this restriction were not imposed, all of these divisions would work.

**Common Variables**
`N` : The numerator in each of these division problems.

`D` : Any number of specific digits in

`N`.

`S` : A sum of digits.

**Two**:

To check if a number `N` is divisible by two, look at `N`'s last digit, `D`. If `D` is divisible by two, then `N` is divisible by two.

**Three**:

To check if a number `N` is divisible by three, find the sum `S` of `N`'s digits. If `S` is divisible by three, then `N` is divisible by three. Repeat as necessary.

**Four**:

To check if a number `N` is divisible by four, look at `N`'s last two digits, `D`. If `D` is divisible by four, then `N` is divisible by four.

**Five**:

To check if a number `N` is divisible by five, look at `N`'s last digit, `D`. If `D` is a five or a zero, then `N` is divisible by five.

**Six**:

To check if a number `N` is divisible by six, check if `N` is divisible by both two and three. If it is, then `N` is divisible by six.

**Seven**:

To check if a number `N` is divisible by seven, look at `N`'s last digit, `D`. Set `N` equal to (`N`-`D`)/10. In essence you are just removing `N`'s final digit. Set `D` equal to 2*`D`. Set `N` equal to `N`-`D`. Check if `N` can be divided by seven. Repeat as necessary.

**Eight**:

To check if a number `N` is divisible by eight, look at `N`'s last three digits, `D`. If `D` is divisible by eight, then `N` is divisible by eight.

**Nine**:

To check if a number `N` is divisible by nine, find the sum `S` of `N`'s digits. If `S` is divisible by nine, then `N` is divisible by nine. Repeat as necessary.

**Ten**:

To check if a number `N` is divisible by ten, look at `N`'s last digit, `D`. If `D` is a zero, then `N` is divisible by ten.

**Eleven**:

To check if a number `N` is divisible by eleven, set `E` equal to the sum of the even (second, fourth, etc.) digits of `N`. Set `O` equal to the sum of the odd (first, third, etc.) digits of `N`. If the absolute value of (`E`-`O`) is eleven or zero, then `N` is divisible by eleven.

**Twelve**:

To check if a number `N` is divisible by twelve, check if `N` is divisible by both three and four. If it is, then `N` is divisible by twelve.

**Thirteen**:

To check if a number `N` is divisible by thirteen, look at `N`'s last digit, `D`. Set `N` equal to (`N`-`D`)/10. In essence you are just removing `N`'s final digit. Set `D` equal to 9*`D`. Set `N` equal to `N`-`D`. Check if `N` can be divided by thirteen. Repeat as necessary.

**Fourteen**:

To check if a number `N` is divisible by fourteen, check if `N` is divisible by both two and seven. If it is, then `N` is divisible by fourteen.

**Fifteen**:

To check if a number `N` is divisible by fifteen, check if `N` is divisible by both three and five. If it is, then `N` is divisible by fifteen.

**Sixteen**:

To check if a number `N` is divisible by sixteen, look at `N`'s last four digits, `D`. If `D` is divisible by sixteen, then `N` is
divisible by sixteen.

**Powers of Two**:

To check if a number `N` is divisible by 2^{X}, look at `N`'s last `X` digits, `D`. If `D` is divisible by 2^{X}, then `N` is divisible by 2^{X}. You can simplify this by dividing both `N` and 2^{X} by two as many times as necessary.

**Composite Numbers**:

To check if a number `N` is divisible by `A``B`, **where **`A` and `B` are coprime (meaning they share no common prime factors), check if `N` is divisible by both `A` and `B`. If it is, then `N` is divisible by `A``B`.