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