Will a number divide by 11, 7 or 13?

Here are three little tricks which could save you a lot of grief
if you're a tough person who doesn't use calculators.
The 11 test The 7 test The 13 test

Will a number divide by 11?

There's a quick way to know if a number will divide exactly by 11 with no remainder - and not many people know it!

What you do:

Put (or imagine) alternate + and - signs in front of the digits
Total them all up
If the total is zero, or if the total divides exactly by 11, then the original number will divide by 11.

Let's try one...91825074 Yuk! But will it divide by 11?

Put the signs in: +9 -1 +8 -2 +5 -0 +7 -4
Get all the "+" bits and "-" bits together if you like: +9 +8 +5 +7 - 1 -2 -0 -4
This becomes: 29 - 7 = 22
22 divides by 11, so that means 91825074 will divide by 11 too!

If you try 17483103
You get +1 -7 +4 -8 +3 -1 +0 -3 which makes -11. That still counts as dividing exactly by 11, so 17483103 divides by 11.
What use is all this?
If you get around to breaking massive numbers into prime factors (this comes up in "The Mean and Vulgar Bits") this trick can save a lot of time!
Another time this comes in handy is when you're the all supreme ruler of a massive planet and the entire population wants to know if they can all be divided neatly up into football teams.
The 11 TEST (With a pretty counter!)

Will a number divide by 7?

On page 111 of "Murderous Maths" you'll find a the long way of testing if a number will divide by 7. However since the book came out, we've found a slightly quicker way!

Take off the last digit and multiply it by 2
Subtract it from the digits you have left.
If the answer divides by 7 (or is zero), then your original number will also divide by 7!

We'll try 161623 Well, there's no point in just testing a little number is there?

Take off the last digit and x 2, so we get 3 x 2 = 6
Subtract this from the other digits so we get 16162 - 6 = 16156. But does this divide by 7?
Do it again! We get 1615 - (6 x 2) = 1615 - 12 = 1603. Does this divide by 7?
On we go... 160 - 6 = 154
...and on... 15 - 8 = 7 YES! 7 divides by 7, so 161623 also divides by 7!

If your number has three or more digits, you could also try this brilliant "7 test" that was sent to us by a Murderous Maths fan in Kent called Jaimie Oldham.

Take off the last two digits then multiply the other digits by 2.
Add your answer to the two digits you took off.
If the answer divides by 7 then your original number will also divide by 7!

Try 2723

Take off the last two digits to leave 27 and then 27 x 2 = 54
Add your answer to the two digits you took off: 54 + 23 = 77.
As 77 does divide by 7 then so does 2723.

When you do any "7" tests, it's handy to remember that 77, 84, 91 and 98 all divide by 7.

Will a number divide by 13?

This is similar to seeing if a number will divide by 7.

Take off the last digit and multiply it by 9
Subtract it from the digits you have left.
If the answer divides by 13 (or is zero), then your original number will also divide by 13!

We'll try 224082

Take off the last digit and x 9, so we get 2 x 9 = 18
Subtract this from the other digits so we get 22408 - 18 = 22390. But does this divide by 13?
A BIT OF EXCITEMENT! As the last digit is "0" we can just knock it off and continue testing. So does 2239 divide by 13?
We get 223 - (9 x 9) = 223 - 81 = 142. Does this divide by 13?
On we go... 14 - 18 = -4. -4 does not divide exactly by 13, so neither does 224082.

However, if we'd started with 224081 the sums would have been...
22408 -9 =22399 then 2239 -81= 2158 then 215 -72= 143 and finally 14 -27= -13
... and as -13 does divide exactly by 13, therefore 224081 divides by 13 too!

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