### Counting Factors

Is there an efficient way to work out how many factors a large number has?

### Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

### Helen's Conjecture

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

# Elevenses

##### Stage: 3 Challenge Level:

In the grid below, look for pairs of numbers that add up to a multiple of 11.

 9 46 79 13 64 90 2 97 25 31 20 22 4 52 55 7

Are there any numbers that can only have one partner?
Are there any numbers that could have more than one partner?
What is special about numbers which have the same set of partners?

Can you find every possible pair?
How can you be sure you haven't missed any?

You may have solved the problem by looking at how close each number is to a multiple of 11...

Here is another grid.
This time, look for pairs that add up to a multiple of 13.

 11 54 93 15 76 106 2 115 29 37 24 26 4 62 65 9

How can you use your insights from the first problem to be sure you have found all the possible pairings?

Thank you to Susanne Mallett from Comberton Village College for introducing us to this problem.