Divisibility by 7
When you learned the various divisibility rules you were probably told that there was no divisibility rule for 7. This is not true! There are many divisibility rules for 7! They're usually a lot more work than dividing by 7, but much more fun!
Here is one way to tell if a number is divisible by 7: Take the ones digit off, double it, and subtract it from the rest of the number. If the result is divisible by 7, so was the original number. You can repeat this process until the result is clearly divisible (or not) by 7.
For example: Take a number like 408254, which is divisible by 7. Take the 4 off the end, and double it: 8. 40825 - 8 = 40817. It's not obvious that this is divisible by 7, so repeat the process. 4081 - 14 = 4067. Again, not obvious. So 406 - 14 = 392. One more time. 39 - 4 = 35. That is divisible by 7!
So why does this work?
Any whole number N can be written in the form N = 10a + b, where b is its final digit. Clearly 21b is a multiple of seven and N is a multiple of seven only if N - 21b is also. But note:
But a - 2b is the result of deleting the last digit from N and subtracting twice that digit from what remains!
Finally: Why choose the number 21 at the beginning of this discussion? It is the first multiple of seven with final digit one, so subtracting 21b deletes the single b in 10a + b.
Here is one way to tell if a number is divisible by 7: Take the ones digit off, double it, and subtract it from the rest of the number. If the result is divisible by 7, so was the original number. You can repeat this process until the result is clearly divisible (or not) by 7.
For example: Take a number like 408254, which is divisible by 7. Take the 4 off the end, and double it: 8. 40825 - 8 = 40817. It's not obvious that this is divisible by 7, so repeat the process. 4081 - 14 = 4067. Again, not obvious. So 406 - 14 = 392. One more time. 39 - 4 = 35. That is divisible by 7!
So why does this work?
Any whole number N can be written in the form N = 10a + b, where b is its final digit. Clearly 21b is a multiple of seven and N is a multiple of seven only if N - 21b is also. But note:
N - 21b = 10a + b - 21b = 10a - 20b = 10 (a - 2b)and 10(a-2b) is a multiple of seven only if a - 2b is a multiple of seven, since 10 has no common factor with seven. So N = 10a + b is a multiple of seven only if a - 2b is a multiple of seven.
But a - 2b is the result of deleting the last digit from N and subtracting twice that digit from what remains!
Finally: Why choose the number 21 at the beginning of this discussion? It is the first multiple of seven with final digit one, so subtracting 21b deletes the single b in 10a + b.
