Before we get to the main event, which is the divisibility rule for 7, let’s first
do a quick review of the rules for 2 through 11.
We all know that a number is divisible by 2 if it’s last digit is even.
For example, 138 ends in an 8, which is even, so 138 is divisible by 2.
For 3 we just check that the sum of the digits is a multiple of 3.
Let’s test it on 777.
Yup, the sum of the digits is 21, a multiple of 3, so 777 is divisible by 3.
Next up is four.
We just check that the last two digits of this number are a multiple of 4.
If they are, so is the original number.
Divisibility by five is another easy one.
The final digit has to be either a zero or a five.
So both these numbers are good to go!
Since 6 is 2 times 3, we need to satisfy the divisibility rule for 2 and 3.
Let’s test it on 2,442.
It ends in a 2, so it’s even and the sum of the digits is a multiple of 3, so 2442
is divisible by 6.
The rule for 8 is just like 4, but we check that the last 3 digits are divisible by 8.
In this number the last 3 digits are 104 which is indeed divisible by 8, so the number is
as well.
Nine works similar to 3 except with nine make sure the sum of the digits is a multiple of
9.
This number has 9 ones, which add up to 9, so it’s divisible by 9.
I think everyone knows ten.
Just make sure the number ends in a zero like this one and you’re guaranteed it’s divisible
by 10.
The rule for 11 is pretty cool and probably the least known of the ones we’ve looked
at so far.
Let’s test it on 253.
Just put alternating signs between the digits.
If the resulting expression adds up to zero or a multiple of 11, guess what – the number
is divisible by 11! Ok.
Now let’s look at 7.
This one is the most unusual of all the rules so far because we can’t just tell from a
glance or a single calculation if it’s divisible by 7 or not.
Here’s how it works.
Let’s take a four-digit number as an example, say 1,876.
First remove the last digit from the number and double it.
So 6 doubled is 12.
Subtract 12 from the remaining digits.
You get 175.
If you’re not sure if 175 is divisible by 7, repeat the same process.
Remove the 5 since it’s the last digit, double it to get 10, and subtract 10 from
the remaining digits.
17 minus 10 is 7 and that is obviously divisible by 7.
Which means so is 1876!
Amazing, right?
But it seems like more of a magic trick, so how on earth does this actually work?
Let’s start by rewriting the number we just used, 1876, as powers of ten.
We would have 1 times 10 cubed plus 8 times 10 squared plus 7 times 10 plus 6.
Now let’s extend that to a general number whose digits are represented by first digit
alpha, second digit beta, and so on all the way to second to last digit phi and last digit
omega.
I put the bar on top to remind us that this is just a string of digits, not a product
of digits.
So we would have alpha times 10 to the nth power plus beta times 10 to the n minus 1
power, and so on, all the way to phi times 10 plus the units digit omega.
Now let’s set this number equal to a multiple of 7 since that’s what we’re investigating
here.
Next I’m going to do something that will seem utterly pointless.
I’m going add 20 omega and subtract 20 omega from the left side.
But actually it’s a clever trick.
Omega and 20 omega combine to get us 21 omega which I’m now going to move to the other
side.
When I do, notice that the right side is still a multiple of 7 which we want to maintain
in order to find out when the left side is also a multiple of 7.
Now let’s factor out a 10 from the left side.
Notice that all my exponents just get reduced by one.
Now here’s the cool part that shows how divisibility by 7 really works.
We know the right side is a multiple of 7 since I can rewrite it as 7 times k minus
3 omega.
What about the left side?
Well, 10 isn’t a multiple of 7, so the only way the number can be a multiple of 7 is if
the remaining factor boxed in red is a multiple of 7.
And notice that 2 omega is twice the final digit which we subtract from the remaining
digits.
We can repeat the process as often as we like since we can keep factoring out 10 and be
left with a number one digit less than before.
We stop the process when we know if the remaining digits are divisible by 7 or not.
One final question for you to ponder.
Among the primes we tested for divisibility rules, why does 7 behave like this?
After all, 2 is prime and it’s a piece of cake.
3 is prime and it also has a simple rule.
Same with 5 and 11.
So why do things change so drastically with 7 and require a process to determine divisibility
rather than a single observation or calculation?
My guess is that it might have something to do with the nature of 7 when you look at its
somewhat more complex and chaotic decimal representation.
But that’s just a guess and I’d love for one of the number theorists among us to give
a more definitive reason.
Let me know in the comments below if you know some other cool divisibility tricks.
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and some good old fun.
Thanks for watching and see you soon!