How can you quickly tell if a number is divisible by 7 and why this trick works

Photo by woodleywonderworks

This trick isn’t as effective as the test for divisibility by 3. It takes more steps:

1. The number is made up of a series of digits: ABCDE, take the last digit off the number. In this case we have ABCD and E.
2. Double the last digit, 2 times E
3. Subtract 2E from the rest of the number, ABCD – 2E
4. Is ABCD – 2E easily recognized as divisible by 7? If not, take your new number and restart at step 1.

Lets try this with an actual number:

1. Is 12345 divisible by 7? 1234 and 5
2. 2 times 5 = 10
3. 1234 – 10 = 1224
4. Is 1224 divisible by 7? Not sure, so I start at step 1. again.
1. Is 1224 divisible by 7? 122 and 4
2. 2 times 4 is 8
3. 122 – 8 = 114
4. Is 114 divisible by 7? Not sure, so I start at step 1. again.
1. Is 114 divisible by 7? 11 and 4
2. 2 times 4 is 8
3. 11 – 8 = 3
4. Is 3 divisible by 7? NO, so 12345 is not divisible by 7.

Another number to try:

1. Is 12341 divisible by 7? 1234 and 1
2. 2 times 1 = 2
3. 1234 – 2 = 1232
4. Is 1232 divisible by 7? Not sure go to step 1.
1. Is 1232 divisible by 7? 123 and 2
2. 2 times 2 equals 4
3. 123 – 4 = 119
4. Is 119 divisible by 7? Not sure, so go to step 1 again.
1. Is 119 divisible by 7? 11 and 9
2. 2 * 9 = 18
3. 11 – 18 = -7
4. Negative 7 is divisible by 7 so 12341 is also divisible by 7.

Why does this work?

• ABCDE = 10 times ABCD + E
• If a number is divisible by 7 and you multiply by another number, your new larger number is still divisible by 7. You just now have another factor. 21 is divisible by 7. 2 times 21 = 42 is still divisible by 7. 42 = 7 * 3 * 2. 2 is our new factor
• ABCDE * 2 = 20 times ABCD + 2E (and still divisible by 7 if it was before)
• If you take a number that is divisible by 7 and add or subtract 7 or a multiple of 7, you have a new number that is also divisible by 7. 14 is divisible by 7, so is 14 + 7, as is 21 + 14, as is 35 minus 7, as is 35 minus (7 times any number), as is 35 minus (21 times any number)
• If you take a number that is divisible by 7 and subtract 21 times any number, you have a new number that is also divisible by 7.
• ABCDE * 2 = (2o * ABCD )+ 2E ; Subtract (21 * ABCD) from both sides and we get a new number with the same divisibility by 7 = 20ABCD + 2E – 21ABCD = 2E – ABCD
• If 2E – ABCD is divisible by 7 then so is ABCD – 2E ; Multiplying by another number, even negative one, does not change divisibility by 7.

So ABCD – 2E makes a smaller number that is easier to test for divisibility by 7, and now we can see why it works. Also notice that ABCD can be any number of digits. Nothing I did above notices how many digits we are working with.