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Print Numerical / Quantitative Reasoning Best Practices #2: The Secret of Number Division

Many numerical problems leave you with a calculation to be done to get to a final answer. This can always be done with a calculator but, strangely enough, they can often be done faster in your head if only you know how. I don’t expect you to become mental arithmetic savants overnight; that is not the aim of this article but there are a few simple things that can be done to speed things up.

Suppose you need to calculate the average speed of a car that has travelled 319 miles in 5h 30m. The way to do this is to divide the distance by the time but, on the face of it, that looks like quite a difficult sum. But is it?

We can easily see that 5h 30m is the same as 5.5 hours which is 1.1*5 or 11*0.5. So, if 319 is divisible by 5 or 11 then the sum is quite easy. Now with a number like 319 it’s not too difficult to test whether it is divisible by 5 or 11 but what if it were a large number? Could we still do it?


Divisible by 2

A number is divisible by 2 if the last digit is even. It doesn’t matter how many other digits are in the number, odd or even, only the last digit matters.

Divisible by 3

Add all the digits of the number together and if the total is a multiple of 3 then the original number is also. Eg. 12345 is divisible by 3 because 1+2+3+4+5=15 => 1+5=6 which is a multiple of 3.

Divisible by 4

If the last two digits are divisible by 4 then the whole number is.Eg. 139742 is not divisible by 4 because 42 is not divisible by 4.

Divisible by 5

If the last digit is 0 or 5 then the number is divisible by 5.

Divisible by 6

Use the tests for two and three. Eg. 139742 is divisible by 2 because the last digit is even but is not divisible by 3 because 1+3+9+7+4+2=26 => 2+6=8 which is not divisible by 3 so it is not divisible by 6.

Divisible by 7

I have encountered many tests for divisibility by 7 but every one of them has been more difficult than the actual division and so renders it pointless.

Divisible by 11

This is the interesting one. Is 139742 divisible by 11? I can say instantly that it is not. Add all the alternate digits together and then the other alternate digits and subtract one from the other. If the answer is 0 or a multiple of 11 then the whole number is divisible by 11. In our example, 1+9+4=14 and 3+7+2=12 which is a difference of 2 so it is not divisible by 11.

At the beginning of the article we had the sum 319/5.5 and now we can see instantly that 319 is divisible by 11 [3+9-1=11] and so we can reduce the fraction to 29/0.5 which is the same as 29*2 which equals 58.

Just 10 minutes’ practise and you’ll be doing that faster than your calculator.

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