Fluency, reasoning and problem solving are at the heart of the new National Curriculum.

This month's maths challenge gets children to work with abunant numbers and perfect numbers to develop these skills.

**Fluency:**Challenges look at factors which require a certain amount of fluency in multiplication tables.**Reasoning**: Children will have to follow a line of enquiry, make generalisations and justify or prove their generalisations using mathematical language.**Problem solving:**Children will need to apply their mathematics to a non-routine problem.

__Challenge 1__

__Have you heard of abundant numbers?__

An abundant number is sometimes called an excessive number. It’s an integer that is less than the sum of its factors (not including the number itself). Other numbers are greater than or sometimes equal to the sum of their factors.

*For example, 12 is the first abundant number. Its factors are 1, 2, 3, 4, 6 and 12. If we discount 12, the sum of the other factors is 16 which is greater than twelve.*

*15 isn’t an abundant number. Its factors are 1, 3, 5 and 15. If we discount 15, the sum of the other factors is 9 which is less than fifteen.*

__Your Classroom Tasks__

**Question 1.** Ask the children to find the other abundant numbers to 100. What do they notice about all of them?

**Question 2.** Can they work out the lowest odd abundant number?

**Question 3. **You could ask them to work out if certain numbers are abundant such as square, prime and triangular numbers. You could ask them to work out why these are either abundant or not or a mixture.

**Question 4.** Can they work out any generalisations and prove them?

__Challenge 2__

__Have you heard of perfect numbers?__

*For example, A perfect number is a positive integer that is equal to the sum of its factors.*

__Your Classroom Task__

The first perfect number is 6. Its factors are 1, 2, 3 and 6. If we discount 6, the sum of the other factors is 6.

**Question 1.** There aren’t many of these, but there is one below 100. Can the children find out which one it is?

**Question 2.** There is one 3-digit perfect number. Can they work out which one this is?

**Question 3**. There is one 4-digit perfect number. Can they work out which one it is?

The next perfect number is in the millions, and would take an awful long time to find!!

Please share these challenges with other teachers if you think they'd find them helpful and don't forget to let us know how you get on!