*Contributed by Mickey Sheu of Naperville North High
School. Edited by Ian Yap and Abe Hassan*

- Take the last digit off of the number.
- Double this digit.
- Subtract this new number from the rest of the original number.
- Keep repeating until you get to a point when you cannot go further.
- If this number is divisible by seven, then so is the original number.

*Example:*

- Number: 12345
- Take off 5, double it to get 10, and subtract: 1234 - 10
- Result: 1224
- Take off 4, double it to get 8, and subtract: 122 - 8
- Result: 114
- Take off 4, double it to get 8, and subtract: 11 - 4
- Result: 3
- Since three is obviously indivisible by seven, 12345 is also not divisible by seven.

*Proof: *All you are doing is subtracting 21*the last digit, preserving the remainder.

- Take the last digit off of the number.
- Subtract this number from the rest of the original number.
- Keep repeating until you get to a point when you cannot go further.
- If this number is divisible by seven, then so is the original number.

*Example:*

- Number: 12345
- Take off 5 and subtract: 1234 - 5
- Result: 1229
- Take off 9 and subtract: 122 - 9
- Result: 113
- Take off 3 and subtract: 11 - 3
- Result: 9
- Since nine is obviously indivisible by eleven, 12345 is also not divisible by eleven.

*Proof:* All you are doing is subtracting 11*the last digit, preserving the remainder.

*13 is trickier. It is so weird that it is unbelievable.*

- Divide the number into groups of three digits, starting from the right.
- Add the first group, subtract the second, add the third, subtract the fourth, etc.
- Keep repeating until you use all of the groups.
- If the result is divisible by thirteen, so is the original number.

*Example 1:*

- Number: 12,345,678,901,234
- Divide into groups of three: 12; 345; 678; 901; 234
- Alternate adding and subtracting: 12 - 345 + 678 - 901 + 234
- Result: -322
- Since -322 is indivisible by thirteen, neither is the original number.

*Example 2: *Let's try a number that is divisible by 13: 13*1532 is 19916.

- Number: 19916
- Divide into groups of three: 19; 916
- Alternate adding and subtracting: 19 - 916
- Result: -897, which is equal to 13*-69
- Therefore, 19916 is divisible by thirteen.

*Note from the contributor: *Hmmm, I cannot find the divisibility test. It's
out there though, because my summer course counselor has showed it to me. Something
about factoring 10^{(3(2N+1))}+1 and 10^{(3(2N))}. I'll keep looking.

*A divisibility test for 17 is usually useless.*

- Take the last digit off of the number.
- Multiply this number by five.
- Subtract this new number from the rest of the number.
- Repeat until you get to a point where you can not proceed any further.
- If this result is divisible by seventeen, then so is the original number.

*Example:*

- Number: 12345
- Take off the 5, multiply by 5, and subtract: 1234 - 25
- Result: 1209
- Take off the 9, multiply by 5, and subtract: 120 - 45
- Result: 75
- Since 75 is not divisible by seventeen, neither is 12345.