Mental Arithmetic Tips #1

When I was at school, calculators hadn’t been invented, so we learned to do maths the old-school way, literally.

We had to do all of our calculations by hand, using pen and paper, and learned how to do multiplication and long division and who knows how many other methods – it was a long time ago now and much of what I learned I probably take for granted these days.

And as part of all this, we also did mental arithmetic – doing math in your head – a dying art, apparently, which is why this article is going to teach you a few useful tricks to help you with some mathematics.

It was actually written by a friend of mine, Brian, who is a fellow magician and who has been a math teacher for most of his life.

Let’s get started.

Validating Calculations

With calculators being so readily available nowadays, it is hardly surprising that many people will turn to these to carry out calculations.

This is all well and good in itself, but it is often useful to know whether the answer obtained from a calculator appears to be correct. This is not because calculators make mistakes – but the people using them do. 🙂

Carrying out approximate calculations or other checks in your head may give you an idea as to whether the calculator answer is correct.

For young people and adults alike, it is highly advantageous to know your multiplication tables, ideally up to 12 × 12. [Editor’s Note: I was bribed by my father to learn these when I was very young – the reward being a Winston Churchill crown, which was a coin worth five shillings at the time, which would be the equivalent of 25 pence these days.]

So, for example, if you were to require the answer to 4.1 × 11.96, we’d expect the answer to be close to 4 × 12 (which is 48).

If we obtained the answer of 490.36 on a calculator, then having done this brief check we’d know that this was the incorrect answer, the actual answer being 49.036.

If we multiply two whole numbers, we can always ascertain the last digit of our answer by multiplying the last pair of digits:

• The answer to 46 × 78 must end in an 8, because the answer to 6 × 8 ends in an 8.
• Similarly, the answer to 47 × 78 must end in a 6, because the answer to 7 × 8 ends in a 6.

This works when multiplying two whole numbers with any number of digits. The answer to 15,423 × 6,147 ends in 1 because the answer to 3 × 7 also ends in 1.

Whilst it doesn’t eliminate the chance of an error when using a calculator, this useful validation is quick and easy to carry out.

Tests for Divisibility

Most people probably know how to quickly look at a whole number and ascertain whether it can be divided exactly by:

• 2, because it ends in 0, 2, 4, 6, or 8
• 5, because it ends in 0 or 5
• 10, because it ends in 0

However there are also fairly easy tests for divisibility for some other numbers.

Divisibility By 3

We can easily see if a whole number can be divided exactly by 3 by adding the digits in that number. If the sum we obtain by adding the digits is a multiple of 3, then the original number is divisible by 3.

Let’s take, for example, the number 72,651. If we add the digits, we get 7 + 2 + 6 + 5 + 1 = 21. 21 is divisible by 3, so 72,651 is divisible by 3.

If we add the digits in the number 648,932, we get 6 + 4 + 8 + 9 + 3 + 2 = 32. 32 is not divisible by 3, and so 648,932 is not divisible by 3.

Let’s look at a very large number. We’ll choose 736,482,736,478,292. Adding the digits in this number gives 7 + 3 + 6 + 4 + 8 + 2 + 7 + 3 + 6 + 4 + 7 + 8 + 2 + 9 + 2 = 78.

Now, let’s suppose we are not sure whether or not 78 is a multiple of 3. All we have to do is repeat the test: 7 + 8 = 15.

15 is a multiple of 3, and therefore 78 is a multiple of 3, and so our original number 736,482,736,478,291 is also a multiple of 3.

We can actually take this test a step further.

If a number is a multiple of 3, then adding 3, 6, or 9 to the number will give another number which is a multiple of 3.

Based on this, when we are adding the digits in a number to check for divisibility by 3, we can simply ignore any digits which are 3, 6 or 9.

For example, we will check to see whether 71,365,938 is divisible by 3. This time we will add all the digits other than any 3, 6, or 9, which gives us 7 + 1 + 5 + 8 = 21. 21 is divisible by 3, and so 713,65,938 is also divisible by 3.

Divisibility By 9

The above test also works for dividing by 9.

If we add the digits in a whole number and the answer is a multiple of 9, then the original number is also a multiple of 9.

And extending this test, as above, when we add the digits, we can ignore any 9s.

For example, if we add the digits in the number 895,931,991, but ignoring any 9s, we have 8 + 5 + 3 + 1 + 1 = 18. As 18 is a multiple of 9, then 895,931,991 is also a multiple of 9.

Divisibility By 4

Checking to see if a number is divisible by 4 involves looking at the last two digits of the number.

Obviously, the number has to be an even number, so we know instantly that if it ends in 1, 3, 5, 7, or 9 that it is not divisible by 4 (or any other even number, in fact).

If the last two digits of the number make a number which is divisible by 4, then the original number is divisible by 4.

For example, the number 71,283,724 ends in 24. 24 is a multiple of 4, so 71,283,724 is divisible by 4. Consequently, any whole number ending in 24 is divisible by 4.

Divisibility By 6

For a number to be divisible by 6, it would have to be both an even number, and divisible by 3.

So to check if a number is divisible by 6, we simply make sure that it’s even, and then carry out the test for divisibility by 3.

The number 34,812 is an even number, and if we add the digits (ignoring the 3, as per the short-cut tip above), we get 4 + 8 + 1 + 2 = 15. 15 is divisible by 3, and so 34,812 is divisible by 6.

Divisibility By 8

A test for divisibility by 8 exists, and in this case we need to look at the last three digits of the number.

If the last three digits of the number form a number which is a multiple of 8, then the original number is also divisible by 8.

This test may require a small division, but numbers have to be even to be divisible by 8, so we only need carry out the test if our original number is even.

For example, 7,862,264 ends in 264. 264 is divisible by 8 (as 33 × 8 = 264). Hence 7,862,264 is also divisible by 8.

Divisibility By 7

Finally, to see if a number is divisible by 7, it’s actually easier to carry out the division itself. 🙂

Conclusion

I hope you are inspired to look at some numbers and see what they are divisible by, and to realize that everyday math need not be as difficult or daunting as you might think.

Many thanks to Brian for writing this helpful article. It even made me remember a tip or two that I’d forgotten about.

Who knows, I may even persuade him to write a few more in future. 🙂

Additional Resources

Brian didn’t know I was going to do this, but the following is his site, designed for math teachers (but it would also be great for those parents who homeschool), that has a bunch of great resources to help children (and adults too) learn maths and stop being afraid of it:

1. Maths Topics & Mental Starters