While working on a divisibility section for my web site, I across some interesting facts about numbers that, to someone who does not know about these things, would seem like mental arithmetic magic.

In case you don’t know, the divisibility of a number refers to whether a number can be evenly divided by another. For example, 24 is divisible by 2 but 13 is not as it leaves a remainder.

Here are some mental arithmetic magic tricks I have found that you can use to impress your friends, your colleagues, your teachers, . . .

1. if a number is divisible by 3 then so are the numbers based on mixing up the digits of the original number. For example, consider 123 which is divisible by 3. Then 132, 213, 231, 312 and 321 (which are obtained by mixing up the digits 1, 2 and 3 that make up 123) are all divisible by 3. This is called a permutation of the digits of a number. Check it for yourself!

2. to make up a number that is divisible by 4, make up a numberr and tag on the end any 2 digit number divisible by 4. For example, I make up the number 111111111, and now I tag 16 (which is divisible by 4) on the end to get 11111111116. This number is divisible by 4. Check it for yourself!

An interesting trick follows on from this one. The following numbers are all divisible by 4: 116, 1116, 11116, 111116, and so on . . . Not what you would expect!

3. if a number is divisible by 6, then any permutations of its digits will give you a new number divisible by 6 as long as the last digit is even. For example, 1272 is divisible by 6. Permutations of its digits while keeping the last digit even gives me 2172, 2712, 1722, 7122, 7212 which are all divisible by 6. Check it for yourself!

4. to make up a number that is divisible by 8, the process is similar to point 2. above. Make up a number and tag on the end any 3 digit number divisible by 8. For example, I make up the number 777777, and now I tag 016 (which is divisible by 8) on the end to get 777777016. This number is divisible by 8. Check it for yourself!

Another interesting trick follows on from this one. The following numbers are all divisible by 8: 7016, 77016, 777016, and so on . . . Again, not what you would expect!

5. if a number is divisible by 9 then so are the numbers based on mixing up the digits of the original number. For example, consider 189 which is divisible by 9. Then so are 198, 819, 891, 918 and 981. Check it for yourself!

6. if a number is divisible by 11, then permutations of its odd digits and/or its even digits will give you a new number also divisible by 11. For example, consider 154 which is divisible by 11. Then so is 451 (obtained by swapping its first and third digits). Another example, consider 1122 which is divisible by 11. Then so is 1221 (obtained by swapping its second and fourth digits). Check it for yourself!

7. if a number is divisible by 12, then any permutations of its digits (except for the last 2) will give you new numbers also divisible by 12. For example, 14652 is divisible by 12. Then so are 16452, 41652, 46152, 61452 and 64152. Check it for yourself!

You have to agree that such tricks do look like arithmetic magic which you can do in your head. Hence the title of this article being ‘mental arithmetic magic’. In case you are wondering ‘why is it so?’ The answer lies in the test that determines whether a number is divisible by another.

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