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Thirty Six Exactly

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

Strange Numbers

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.

Sieve of Eratosthenes

Age 11 to 14
Challenge Level


Why do this problem?

This problem offers students opportunities to explore multiples in more depth than usual, in particular looking at the links between multiples of different numbers.  It also encourages students to see the connection between primes and multiples.


Possible approach

"What are the first few multiples of 2?"
"2, 4, 6, 8, 10, ..."
"And multiples of 7?"
"7, 14, 21, 28, 35, ..."

"Great. We'll be investigating properties of multiples today."
[Hand out sheets of smaller grids, one sheet per pair of students.]
"I'd like you to shade in all the multiples of 2 except 2, but before you do that, turn to your neighbour and try to predict what patterns you'll produce."
[Give them a minute to make predictions and do the shading. Emphasise that there is no need for beautiful shading.]
"I'd like you to shade in all the multiples of 3 except 3. Again, before you do that, turn to your neighbour and try to predict what patterns you'll produce."
[Give them a couple of minutes to do this.]
"Were your predictions correct? Why did you make those predictions?
Can you explain why you get different patterns for multiples of 2 and multiples of 3?"
"Now let's think about what happens when we combine these multiples."
[Hand out master grid, with multiples of 2 already crossed out.]
"We'll use this as our master grid to keep a running record of our findings.  It's already got the multiples of 2 crossed out.  Before you cross out the multiples of 3, can you and your partner predict what will happen?  Will you cross out any numbers that are already crossed out?  If so, which ones?"
[Give them a couple of minutes to work on this, and then ask them to report back.]

"What am I going to ask you to do next?"
"OK, so now explore what happens for multiples of 4, 5, 6 and 7.  Before you shade in the multiples on the small grids, try to predict what patterns might emerge. After you've 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've updated the master grid, try to explain why some numbers have been crossed out again and others haven't."
[Give them a few minutes for this.]
"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?" 

"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?"
[You might want to have some 2-400 grids available in case students would like to try it.]

Key questions

Which numbers get crossed out more than once, and why?

Which numbers don't get crossed out at all, and why?
Which possible factors do we need to consider in order to decide if a number is prime? 


Possible support

By working in pairs we are encouraging students to share ideas and support each other.  


Possible extension

"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?"