The Rule: How to Know if it's Divisible by 7

There is a three-step rule we can follow to determine if any number is divisible by 7. The rule is especially useful for large numbers and when we don't have a calculator. These are the steps we follow, using 1,715 as an example number:

1. Remove the last digit of the number.
   Example: Remove the 5 from the end of 1,715. The truncated number is 171 and the removed number is 5.
   1,715 → 171.
2. Double the removed number and then subtract it from the truncated number.
   Example: 5·2 = 10, then 171 - 10 = 161.
   171 → 161.
3. Repeat steps 1 and 2 until the number is a recognizable number that's divisible by 7.
   Example: We got to 161 after step 2. Let's repeat step 1 and 2. New removed number is 1. New truncated number is 16. 1·2 = 2. 16 - 2 = 14.
   161 → 16 → 14
   We can easily recognize that 14 is a number divisible by 7 (14/7 = 2). Therefore, the original number 1,715 is divisible by 7.

Divisibility Rule for 7 Example Problem

Determine if the number 24,640 is divisible by 7.

Solution:

1. First, let's remove the last digit from 24,640. The removed number is 0 and the new truncated number is 2,464.
   24,640 → 2,464.
2. Since the removed number is 0, multiplying it by 2 and subtracting it from the truncated number will not do anything. Skipping step 2, step 3 tells us to repeat steps 1 and 2.
3. The new removed number is 4 and the new truncated number is 246. Now, 4·2 = 8. 246 - 8 = 238.
   2,464 → 246 → 238
4. Let's repeat steps 1 and 2 once again. The removed number is 8 and the new truncated number is 23. 8·2 = 16. 23 - 16 = 7.
   238 → 23 → 7
5. We recognize 7 as a number divisible by 7 since it is 7 itself! Therefore, the number 24,640 is divisible by 7.