Dividing by 7

Tuesday, January 12, 2010

To find out if a number is divisible by seven, take the last digit, double it, and subtract it from the rest of the number.
Example: If you had 203, you would double the last digit to get six, and subtract that from 20 to get 14. If you get an answer divisible by 7 (including zero), then the original number is divisible by seven. If you don't know the new number's divisibility, you can apply the rule again. 
Matthew Correnti describes this method:

If you do not know if a two-digit number, call it ab, is divisible 
by 7, calculate 2a + 3b. This will yield a smaller number, and if 
you do the process enough times you will eventually -- if the 
number ab is divisible by 7 -- end up with 7.

You can use a similar method if you have a three-digit number abc: 
take the digit a and multiply it by 2, then add it to the number bc, 
giving you 2a + bc; repeat and reduce until you recognize the 
result's divisibility by seven. With a four-digit number abcd, take 
the digit a and multiply by 6, then add 6a to bcd giving. This 
usually gives you a three-digit number; call it xyz. Take that x 
and, as described previously, multiply x by two and add to yz 
(i.e., 2x + yz). Again, repeat and reduce until you recognize the 
result's divisibility by seven.

Another visitor observes:

Here is one formula for seven...

3X + L

L = last digit
X = everything in front of last digit.

All numbers that are divisible by seven have this in common. 
There are no exceptions.

For example, 42: 3(4) + 2 = 14.
Seven divides into 14, so it divides into 42.

Next example, 105: 3(10) + 5 = 35.
Seven divides into 35, so it divides into 105.

Here is another formula for seven:

4X - L

When using this formula, if you get zero, seven or a multiple of seven, 
the number will be divisible by seven.

For example, 56: 4(5) - 6 = 14.
Seven divides into 14, so it divides into 56.

Next example, 168: 16(4) - 8 = 56.
Seven divides into 56, so it divides into 168.

Similarly:

The formula for 2 is 2X + L
The formula for 3 is 4X + L
The formula for 4 is 6X + L
The formula for 5 is 5X + L
The formula for 6 is 2X + L and 4X + L -- in other words, the formulas for 2 and 3
                                          must work before the number is divisible by 6.
The formula for 9 is X + L
The formula for 11 is X - L
The formula for 12 is 2X - L
The formula for 13 is 3X - L
The formula for 14 is 4X - L and 2X + L -- in other words, the formulas for 7 and 2 
                                           must work before the number is divisible by 14.
The formula for 17 is 7X - L
The formula for 21 is X - 2L
The formula for 23 is 3X - 2L
The formula for 31 is X - 3L

Sara Heikali explains this way to test a number with three or more digits for divisibility by seven:

1. Write down just the digits in the tens and ones places.
2. Take the other numbers to the left of those last two digits, 
and multiply them by two.
3. Add the answer from step two to the number from step one.
4. If the sum from step three is divisible by seven, then the 
original number is divisible by seven, as well. If the sum is 
not divisible by seven, then the original number is not 
divisible by seven.

For example, if the number we are testing is 112, then
1. Write down just the digits in the tens and ones places: 12.
2. Take the other numbers to the left of those last two digits, 
and multiply them by two: 1 × 2 = 2.
3. Add the answer from step two to the number from step one: 
12 + 2 = 14.
4. Fourteen is divisible be seven. Therefore, our original 
number, one hundred twelve, is also divisible by seven.