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Adding All Nine

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

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Sieve of Eratosthenes

Stage: 3 Challenge Level: Challenge Level:1

You will need to print one copy of this 2-100 master grid, and a copy of this sheet of smaller grids.

multiples of 2


On the first small grid, shade in all the multiples of 2 except 2.
  • What do you notice?  Can you explain what you see?
  • Now update the master grid, by crossing out the multiples of 2 except 2.




Multiples of 3

On the second small grid, shade in all the multiples of 3 except 3.
  • What do you notice?  Can you explain what you see?
  • Before you update the master grid, can you predict what will happen?  Will you cross out any numbers that are already crossed out?  If so, which ones?
  • Now update the master grid, by crossing out the multiples of 3 except 3.  Can you explain why some numbers have been crossed out twice and others only once?
     

Use the next four small grids to explore what happens for multiples of 4, 5, 6 and 7.
  • Before you shade in the multiples of each number (but not the number itself), try to predict what patterns might emerge.
  • After you have shaded in the multiples, try to explain the patterns you've found.
  • Before you update the master grid, try to predict what will happen.  Will you cross out any numbers that are already crossed out?  If so, which ones?
  • After you have updated the master grid, try to explain why some numbers have been crossed out again and others haven't.
 
Now look at the master grid.  What is special about the numbers that you haven't crossed out?

What would change on the master grid if you were to cross out multiples of larger numbers? 

We're used to working with grids with ten columns, but you might find an interesting result if you use this six-column grid instead.  Can you predict what you will see?  Try it!
 
 
Final challenge
 
Imagine you want to find all the prime numbers up to 400.
You could do this by crossing out multiples in a 2-400 number grid. 
Which multiples will you choose to cross out? 
How can you be sure that you are left with the primes?

(Here is a 2-400 number grid if you want to try it.)
 
 
With thanks to Vicky Neale who created this task in collaboration with NRICH.