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# Sieve of Eratosthenes

*With thanks to Vicky Neale who created this task in collaboration with NRICH.*
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Age 11 to 14

Challenge Level

*Sieve of Eratosthenes printable sheet
Printable grids - small 2-100 grids, 2-100 master grid, six-column grid, 2-400 grid*

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

**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.

**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!

**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.)

**Final challenge**

Imagine you want to find all the prime numbers up to 1000 by crossing out multiples in a 2-1000 number grid.

Which number will you cross out last?

A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

All strange numbers are prime. Every one digit prime number is strange and a number of two or more digits is strange if and only if so are the two numbers obtained from it by omitting either its first or its last digit. Find all strange numbers.