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In this article, I want to show you how similar tricks can be used for other numbers.

Originally I planned to prove how some of these tricks worked, but in talking through the tricks for other numbers the article ended up getting rather large and so I have instead decided to just present the tricks in the article, with the promise of proving some of them in the next article.

In particular, I want to give you some rules for telling whether or not a particular number is divisible by any of the numbers between 1 and 13. Just a word of warning, throughout this article I jump between calling these “division tricks” and “division rules”. To me a rule is something that works in all cases, a trick is something that works sometimes, therefore a trick needs only to be shown to hold some of the time, whereas a rule needs to be proven for all cases. Since what I am about to show you do work all of the time, they are rules. However, since I haven’t yet provided you with proof of them, you should consider them tricks for now. Also, doesn’t calling them tricks add a sense of magic to them!

Let’s start with the easy cases. Every number is divisible by 1, see that was easy. One down, 12 numbers to go.

Similarly, it is easy to tell if a number is divisible by 2, 4, 5, or 10 without any fancy tricks.

For a number to be divisible by 2 it simply needs to be even. To be divisible by 10, the number must end in a zero. To be divisible by 5, the number simply has to end in a zero or a 5.

What about being divisible by 4? Well since 4=2x2, saying a number is divisible by 4 is equivalent to saying a number is divisible by 2 twice. Another way to put this is that a number is divisible by 4 if, when you divide it by 2 it remains even.

As an example, we see that 246 is not divisible by 4 since 246/2= 123 which is odd, but 248 is divisible by 4 since 248/2=124 which is even.

This trick of breaking a number down into its factors and using the divisibility rules for these factors to create a rule for the original number will be very useful for us later.

There is actually an alternative way to check if a number is divisible by 4 which I will describe below.

I will now lay out the rules for divisibility by 3,4,6,7,8,9,11,12 and 13. Some of these are easier than others and so, rather than go through these in numerical order, I have instead grouped them together in terms of similarity, and from easiest to most difficult to remember. Let’s begin.

Divisible by 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

So, for example, 123 is divisible by 3 since 1+2+3=6, which is of course divisible by 3.

What about 123456789?

Well adding up the digits of 123456789, we find that

1+2+3+4+5+6+7+8+9=45.

Perhaps you can tell straight away that yes, 45 is divisible by 3 and so 123456789 is divisible by 3, but suppose you couldn’t or you weren't too sure. Since we are just asking if 45 is divisible by 3 we can apply the trick again. In particular, we could calculate

4+5=9.

It is now much clearer that 45 is divisible by 3 and thus 123456789 is divisible by 3.

It is often very useful to apply a divisibility trick multiple times, especially if you are dealing with large numbers. For example,

is 123456789999999999999 divisible by 3?

Applying the trick once we calculate

1+2+3+4+5+6+7+8+9+9+9+9+9+9+9+9+9+9+9+9+9=153,

which maybe we are not too sure about, so let’s apply it again

1+5+3=9,

so now we are certain that 123456789999999999999 is divisible by 3.

Divisible by 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

This is probably the most well know of these tricks. Using it we see, for example, that 817 is not divisible by 9, since 8+1+7=16, which is not divisible by 9.

We can apply this trick multiple times if needed. So, for example, we see that for the number 99998999979999,

9+9+9+9+8+9+9+9+9+7+9+9+9+9=123

and

1+2+3=6

which clearly is not divisible by 9, and so 99998999979999 is not divisible by 9.

Divisible by 6

A number is divisible by 6 if it is divisible by 2 and 3.

Here, since 6=2x3, we are just going to combine the rules for divisible by 2 and divisible by 3 to form the rule for divisible by 6. Doing this we see that

A number is divisible by 6 if it is even and the sum of its digits is divisible by 3.

Recall we showed above that 123 was divisible by 3, since 1+2+3=6, which is divisible by 3, but clearly, 123 is not even and so 123 is not divisible by 6.

Similarly, 256 is even but 2+5+6=13 which is not divisible by 3, and thus 256 is not divisible by 6.

3024 is however divisible by 6 since it is clearly even and 3+0+2+4=9, which is clearly divisible by 3.

Divisible by 4

A number is divisible by 4 if the two-digit number at its end is divisible by 4.

This rule may seem a little confusing at first, but it is very easy to apply. For example, we see that 8127 is not divisible by 4 since the two-digit number at the end, 27, is not divisible by 4.

Above we saw that 246 was not divisible by 4, since we couldn’t divide it by 2 twice, but here we conclude that it is not divisible by 4 since the two-digit number at the end, 46, is not divisible by 4.

Divisible by 8

A number is divisible by 8 if the three-digit number at its end is divisible by 8.

Note that this trick is really useful in some cases, and not so useful in others. For example, using this it is easy to see the not so obvious fact that

123456789999999999999999888

is divisible by 8, since the three-digit number at the end, 888, is divisible by 8.

However, using this trick it may not be as obvious whether or not the number 1782 is divisible by 8, since it might not be obvious whether or not the three-digit number at the end, 782, is divisible by 8. In these cases, where it is not so obvious, there are a lot of things you can do.

You could, for example, make the point that since 8=2x2x2, a number is only divisible by 8 if it is divisible by 2 three times. Let’s check this, 782/2= 391 which is odd, and so 782 is not divisible by 8 and thus 1782 is not divisible by 8.

Alternatively, you could, for example, pick a number you know to be divisible by 8 which is close to the number you are unsure about. So since we are unsure about 782 we could, for example, pick 800. From here we note that 800–782=18 and so, since the numbers are not a multiple of 8 away from each other, 782 is not divisible by 8 and thus 1782 is not divisible by 8.

Divisible by 12

A number is divisible by 12 if it is divisible by 3 and 4.

Here, since 12=3x4, we are just going to combine the rules for divisible by 3 and divisible by 4 to form the rule for divisible by 12. Doing this we see that

A number is divisible by 12 if the sum of all its digits is divisible by 3 and if the two-digit number at its end is divisible by 4.

So, for example, is 123456789 divisible by 12? No, since the two-digit number at the end, 89, is not divisible by 4.

What about 9898898944? Well, it is divisible by 4, since the two-digit number at the end, 44, is divisible by 4. We just need to check if it is divisible by 3.

9+8+9+8+8+9+8+9+4+4=76

and

7+6=13

which is not divisible by 3 and thus 9898898944 is not divisible by 12.

Divisible by 11

A number is divisible by 11 if the alternating sum of its digits, from left to right, is divisible by 11.

Said in more plain English, if you want to check if a number is divisible by 11, simply start with the first digit on the left, subtract the second digit, add the third digit, subtract the fourth and continue in this manner. If the number you end up with after this calculation is divisible by 11 then so was your original number.

As an example, 15897 is not divisible by 11 since

1–5+8–9+7=2,

which is not divisible by 11.

As another example, which you may remember from my previous article, 123456789666600000006 is divisible by 11 since

1–2+3–4+5–6+7–8+9–6+6–6+6–0+0–0+0–0+0–0+6=11

which is clearly divisible by 11.

What remains are the rules for division by 7 and 13. These are certainly the “most complicated” of the rules, although they are very easy to use in practice.

Divisible by 7

To check if a number is divisible by 7, remove the last digit of the number, double it, and subtract it from the remainder of the number. If this is divisible by 7, then the original number was divisible by 7.

Ok, so let’s launch straight into an example to help you follow this. We know 777 is clearly divisible by 7, but let’s check it with this rule. The last digit of this number is 7, so we remove this, double it to get 14, and we subtract it from the remainder of the number, which is 77. In particular, we have

77–2(7)=77–14=63.

Since 63 is divisible by 7, 777 is divisible by 7.

In case there is still some confusion, let’s write down an algebraic expression for what we are saying. Let a,b,c,d,e,f,g all be numbers between 0 and 9. Then the number abcdefg is divisible by 7 if

abcdef - (2g)

is divisible by 7.

Let’s do another example in which we have to apply the trick more than once.

Is 12345 divisible by 7? First, we calculate,

1234-(2)(5)=1234–10=1224.

At this stage we are still not sure, so let’s apply the rule again and calculate

122-(2)(4)=122–8=114.

Maybe we are still not sure, so let’s apply the rule one last time and calculate

11-(2)(4)=11–8=3.

Since 3 is clearly not divisible by 7 we are now certain that 12345 is not divisible by 7.

Divisible by 13

To check if a number is divisible by 13, remove the last digit of the number, multiply it by 4, and add it to the remainder of the number. If this is divisible by 13, then the original number was divisible by 13.

Ok, so let’s begin this time by giving an algebraic expression for what we are saying. Let a,b,c,d,e,f,g all be numbers between 0 and 9. Then the number abcdefg is divisible by 13 if

abcdef + (4g)

is divisible by 13.

Consider 12345 again. Is it divisible by 13? First, we calculate,

1234+(4)(5)=1234+20=1254.

At this stage we are still not sure, so let’s apply the rule again and calculate

125+(4)(4)=125+16=141.

Maybe we are still not sure, so let’s apply the rule one last time and calculate

14+(4)(1)=14+4=18.

Since 18 is not divisible by 13 we are now certain that 12345 is not divisible by 13.

So there you have it, divisibility tricks for the numbers 1 to 13. To finish up, let me just make two important points:

  1. These are not the only rules/tricks for showing divisibly by these numbers. In fact, there are at least 3 other tricks I know of for checking if a number is divisible by 13. I have just presented the ones that I think are the nicest/easiest to understand.
  2. There are similar tricks for numbers greater than 13, just because I stopped at 13 doesn’t mean the tricks stop. Some of these are quite simple, for example, to be divisible by 14 you need to be divisible by 2 and 7, so you would just combine these rules together. Others however can get very messy and so I decided to stop at 13 but would encourage you to go and try to figure out or look up some others.

With that said, thanks for reading, and please keep an eye out for the next article in which we delve into exactly why these rules exist and how we can prove they will always work.

©Stephen Witha (PhD)
Cantor's paradise