You might like to start by investigating the patterns you get when you shade multiples of a number on a grid. Download here

For the prime sieve investigation use a 100 square grid (download here):

Circle the number 2. Put a line through every multiple of 2 up to 100.

Circle the number 3. Put a line through every multiple of 3 up to 100.

Circle the number 5. Put a line through every multiple of 5 up to 100.

Circle the number 7. Put a line through every multiple of 7 up to 100.

What do you notice about the multiples of 8, 9 and 10?

What can you say about the numbers that are not crossed out?

4 has 3 factors: 1, 2 and 4. How many factors does 5 have? What about the other circled numbers?

Put a circle about all the other numbers that are not crossed out except the number 1. How many factors do these numbers have? What can you say about them?

For more investigations download and try a 1 to 36 grid or a 1- 400 grid.

What do you notice about these coloured circles?

Click here to go to the brilliant challenge called Xavi’s T-shirt on the NRICH website.

Also try the interactivity for Xavi’s scarf. Given these coloured circles (in new colours) can you work out which numbers they represent?