Write out a list of all the numbers from 1 to, say, 100 (it may help to put them in a rectangle; depending on the width of the rectangle you may see pretty patterns. Cross out all the multiples of 2, then all the multiples of 3; 4 is crossed out already (by 2) so move on to 5 and cross out all its multiples. Then 6 is crossed out already (by 2 and 3), so move on to 7, and so on. Put a circle around 1, because it’s special (mathematicians call it a “unit”). Everything that is left is a “prime number”. Write a list of them.

Now go back to the numbers you have crossed out (the “composite” numbers). The first of these was 4; 4 = 2 × 2, so you can write 4 as a product of prime numbers. 6 = 2 × 3; 8=2 × 2 × 2 (notice that it’s fine to use more than two prime numbers); 9 = 3 × 3, and so on. Go through the composite numbers, trying to find a set of prime numbers that multiply together to give each of them. We call such a list a “prime factorisation”.

What do you think the prime factorisation of a prime number is?

Can you find a number that doesn’t have a prime factorisation?

Can you find a number with two different prime factorisations (or, equivalently, can you find two different lists of prime numbers that multiply together to give the same value? Reorderings of the same list, like 2 × 3 = 3 × 2, don’t count).

Pick your favourite two numbers (small ones are going to be easier). Work out their prime factorisations. Multiply them together. Work out the prime factorisation of the product. Do this for a few other pairs. What do you notice?

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