Times Tables and the Fundamental Theorem of Arithmetic

Sounds very posh, doesn’t it? ‘Fundamental Theorem of Arithmetic’.

What on earth does it mean?

By now, you should be able to calculate the prime factors of a number quite easily. If you can’t, please go back to the Prime Numbers (2) task and have a go at that. This is another example where a good knowledge of times tables comes in very useful.

Take a number such as 420 and work out its prime factors (remember that 1 is not a prime number).

We see that the prime factors are 3 x 7 x 2 x 2 x 5, which we can write as 2 x 2 x 3 x 5 x 7.

Again, it comes out to be the same prime factors as before.

This is what the Fundamental Theorem of Arithmetic says: It doesn’t matter how you find the prime factors of a number, the answer is always the same.

This applies to any number, even a number as big as

2 748 591 593 007 252 748 549 960 127 304 262

This is not as obvious as it may seem. You can try it yourself by finding the prime factors of some nice juicy numbers. Find the prime factors in different ways as I have done above using your fantastic knowledge of times tables facts and check they always come out to be the same.

Here are some numbers you might like to try to factorise:

100,   1485,    770,    14 800,    220 500

All these numbers have quite easy prime factors, so all you need to do is keep a cool head and don’t make any mistakes with your dividing. You will need to try at least two different ways of finding the prime factors of each number so that you check the answer is always the same.

No calculators, remember. Good luck!