Divisibility by 13, and a general method for any prime number.

Here's something you already know. 10 = 5 × 2. Because of this well known fact, all we have to do is check the last digit of any number to decide whether a number is divisible by 2 (last digit even) or divisible by 5 (last digit 0 or 5).

For any other prime number, the method for divisibility will be like this.

Step 1: Split the number into two new numbers: the last digit and all the rest.
Step 2: Multiply the last digit by some special number. (This number is dependent on which prime we are using.) Then add this multiplied value to the value we are calling "the rest".
Step 3: Check to see if you can tell if this new value is divisible by the prime in question. If you cannot tell yet, go back to step 1 and continue until you get a number where you can tell.

Examples

Divisibility by 3: The special number we multiply the last digit by is 1. (Nice and easy.)
Divisibility by 7: The special number we multiply the last digit by is 5.
Divisibility by 11:  The special number we multiply the last digit by is 10.
Divisibility by 13: The special number we multiply the last digit by is 4.

(Finding the special number is done by understanding number theory. What we are looking for is the number we multiply by 10 so that the remainder when we divide by the prime is 1.)

Okay, let's look at 5,280, the number of feet in a mile, and see if it is divisible by 3, 7, 11 or 13. (It should be obviously that it is divisible by 2 and divisible by 5.)

Divisible by 3.
1. Split 5280 into 528 and 0. Multiply 0 by 1 and add to 528, which gives us 528. 528 is a little too big, so let's do the method again.
2. Split 528 into 52 and 8. Multiply 8 by 1 and add to 52, which gives us 60. 60 is divisible by 3, so the original number 5,280 is also divisible by 3.

Divisible by 7.
1. Split 5280 into 528 and 0. Multiply 0 by 5 and add to 528, which gives us 528. 528 is a little too big, so let's do the method again.
2. Split 528 into 52 and 8. Multiply 8 by 5 and add to 52, which gives us 92. Still a little to big to see divisibility by 7 easily, so one more step.
3. Split 92 into 9 and 2. Multiply 2 by 5 and add to 9 to get 19.  19 is not divisible by 7, so the original number 5,280 is also not divisible by 7.

Divisible by 11.
1. Split 5280 into 528 and 0. Multiply 0 by 10 and add to 528, which gives us 528. 528 is a little too big, so let's do the method again.
2. Split 528 into 52 and 8. Multiply 8 by 10 and add to 52, which gives us 132. Still too big, so one more step.
3. Split 132 into 13 and 2. Multiply 2 by 10 and add to 13, which gives us 33.  33 is obviously divisible by 11, so 5,280 is also divisible by 11.

Divisible by 13.
1. Split 5280 into 528 and 0. Multiply 0 by 4 and add to 528, which gives us 528. 528 is a little too big, so let's do the method again.
2. Split 528 into 52 and 8. Multiply 8 by 4 and add to 52, which gives us 84. Not crystal clear, so let's go one more step.
3. Split 84 into 8 and 4. Multiply 4 by 4 and add to 8, which gives us 24. 24 is not divisible by 13, since 2 × 13 = 26. This means the original number 5,280 is not divisible by 13.

Tomorrow: Relatively prime.

Is 7,308 divisible by 7?

In math, we can always divide a by b unless b is zero. But when we say "divisible by 3" for example, we mean evenly divisible by 3, that when we divide by 3 we get a whole number and no remainder or decimal part.

At some point in grade school - I don't know the curriculum year by year - students learn about odd and even numbers, and that numbers ending in 0, 2, 4, 6 and 8 are even and can be divided by 2.  Soon after, they learn that numbers ending in a 0 or 5 are divisible by 5.  Here on the blog, we learned the method for divisibility by 3 and the method for divisibility by 9, both dealing with the sum of the digits. The method for 9 was called "casting out nines" way back in the day (19th Century), and the method for 11 was called "casting out elevens".

Armed with this information, we can tell if a number is divisible by several of the smallest prime numbers, 2, 3, 5 and 11.  But another small prime, namely 7, has been skipped over.  Here is a method for seeing if a number is divisible by 7.

Example 1: Let's take 7,308, a number I chose at random. Here is the method as I was taught in grade school.

 
Step 1: Split the number into the ones digit and the rest of the number. In this example, that means 7,308 becomes 730 and 8.

Step 2: Subtract two times the ones digit from the rest of the number. In this case, we have

730
-16
714

Step 3: If you can tell if the new number is divisible by 7, stop. If not, take the new number and go back to Step 1.

In this case, I can look at the number and say "7 is divisible by 7 and 14 is divisible by 7, so 714 will be divisible by 7."  But let's assume that some people who don't use numbers as often as I do still can't tell and continue with our process, splitting the number into 71 and 4.

71
-8
63

I would hope that everyone remembers 63 = 7 × 9, but even if someone didn't, they could split this number into 6 and 3 and continue.

 6
-6
 0

In general, we should say the process ends when we get a number less than 70 and ask the people using the method to be able to "eyeball" numbers less than 70 and tell if 7 goes in evenly. If someone doesn't remember these numbers, we could get a negative answer.

Example 2: Let's say you can't remember if 85 is divisible by 7. (Hint: It's not.) You can split it into 8 and 5, double the 5 to 10 and subtract.

  8
-10
 -2

This method can leave us with a negative number. If you don't like negative numbers, here is another version of the same idea that uses addition instead of subtraction.

Example 1a: Let's again start with 7,308.

Step 1: Split the number into the ones digit and the rest of the number.  In this case, we'll have 730 and 8, just like before.

Step 2: Multiply the ones digit by 5 and add to the rest of the number. In this case, this means adding 730 and 8 × 5 = 40.

730
+40
770

Step 3: If you can tell if the new number is divisible by 7, stop. If not, take the new number and go back to Step 1. I would hope all my students could see 770 is divisible by 7.

Example 2a: This time, let's use 822.  Split it into 82 and 2, multiply 2 by 5 and get 10.

 82
+10
 92

92 is not divisible by 7, but if we aren't sure of that, split it into 9 and 2, multiply 2 by 5 and add again.

  9
+10
 19

The person using this method needs to know 7 does not go into 19. One "drawback" of this second method is that if we have a two digit number, splitting it again can make a bigger two digit number, so "eyeballing" numbers less than 70 is a non-negotiable skill.

Tomorrow: A similar method of divisibility by 13.