# Divisibility Rules

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

### Divisibility By Seven

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

Example:

1. Number: 12345
2. Take off 5, double it to get 10, and subtract: 1234 - 10
3. Result: 1224
4. Take off 4, double it to get 8, and subtract: 122 - 8
5. Result: 114
6. Take off 4, double it to get 8, and subtract: 11 - 4
7. Result: 3
8. 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.

### Divisibility By Eleven

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

Example:

1. Number: 12345
2. Take off 5 and subtract: 1234 - 5
3. Result: 1229
4. Take off 9 and subtract: 122 - 9
5. Result: 113
6. Take off 3 and subtract: 11 - 3
7. Result: 9
8. 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.

### Divisibility By Thirteen

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

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

Example 1:

1. Number: 12,345,678,901,234
2. Divide into groups of three: 12; 345; 678; 901; 234
3. Alternate adding and subtracting: 12 - 345 + 678 - 901 + 234
4. Result: -322
5. 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.

1. Number: 19916
2. Divide into groups of three: 19; 916
3. Alternate adding and subtracting: 19 - 916
4. Result: -897, which is equal to 13*-69
5. 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.

### Divisibility By Seventeen

A divisibility test for 17 is usually useless.

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

Example:

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