Number theory type questions following on from ATM/MA conf 2023
At the Common Factors joint conference, Charlie and Claire discussed the problems:
Below are some problems that use similar ideas to those used in the above problems. There is also a brief description of each.
Number rules ok: 14-16. This problem asks students to find algebraic or pictorial arguments to show that statements about numbers are true (and there is a challenge to find other statements and prove them). There is an example in the Getting Started section.
Take three from five: 11-16. This problem asks if you have a set of 5 numbers, is it always possible to pick three which sum to a multiple of 3. There is a video and an interactivity to accompany the problem. Could also be used as an introduction to modular arithmetic, and the ideas link into Square Remainders.
Three consecutive odd numbers: 11-16. This problem asks students if there are any sets of three consecutive odd numbers where all three are prime (apart from 3, 5, 7). There is a proof sorter to help scaffold a proof.
Adding Odd numbers: 11-16. This problem asks students what happens when they add up the first $n$ odd numbers, and there is a proof sorter to help.
Adding Odd numbers Part 2: 16-18 - this is an alternative proof sorting using induction. For sixth form students it might be worth comparing the method here and in the previous problem.
Where are the primes? 11-16. This problem asks students to show that all prime number must be one more than or one less than a multiple of 6 (with the exception of 2 and 3). A nice follow on from Three consecutive odd numbers, and there are two methods suggested.
What does it all add up to? 11-18. This problem asks students to consider what happens if they add up four consecutive numbers, and then to prove that they always get 2 more than a multiple of 4. There is a proof sorter to help.
Different products: 14-16. This problem also considers four consecutive numbers, but in this case students are asked to consider the difference between the product of the middle two numbers and the product of the first and last number. There is a proof sorter to help.
The following three problems also consider four consecutive numbers, and give several conjectures for students to prove.
- Starting to consider 4 consecutive numbers: 11-16
- Continuing to explore 4 consecutive numbers: 14-16
- More to discover about consecutive numbers?: 14-16
Impossible sums 14-18. This problem asks students to show that every number which is not a power of two can be written as a sum of consecutive numbers, and that every number which is a power of two cannot. There are three proof sorters showing different methods of showing that if a number is a power of 2 then it cannot be written as a sum of consecutive numbers and a proof sorter for the converse (if a number is not a power of two then it can be written as a sum of consecutive numbers. For a version of the problem without the proof sorters see Polite Numbers.
Always Perfect: 14-18. This problem asks students to show that if you take the product of two numbers that differ by 2 and add one you get a perfect square, and then builds up to multiplying four consecutive numbers and adding one, showing that this also gives a perfect square. There are two methods suggested for each part, and a video demonstrating quartic factorisation in the Getting Started section.
Common divisor: 16-18. This problem asks students to find numbers which always divide certain expressions. There is more support here for students in hide and reveal buttons than in the STEP question it is based on (Divisible Factorisations)
Always 2: 14-18. This problem asks students to find three integers which when you multiply two of them together and add the third you always get the answer 2 (and then extend so you always get the answer $k$ - what values of $k$ are possible?).
STEP Questions:
Digital Equation: STEP Mathematics 1, 2003, Q7 - This problem asks students to find all the three digit numbers that satisfy a certain condition.
Square Remainders: STEP Mathematics III, Specimen paper, Q10(i) - This problem asks students to find the reminders when odd and even numbers are divided by 8.
Proper Factors: STEP Mathematics I, 2009, Q1 - This problem involves using prime factorisations to find the number of proper factors.
Divisible Factorisations: STEP Mathematics I, 1996, Q3 - This problem involves factorising algebraic expressions and using number theory to find factors that divide them
Square Difference: STEP Mathematics I, 2014, Q1 - This problem takes the same ideas as in What's Possible, but extends the ideas to consider in how many different ways expressions of the form $pq$ (where $p$ and $q$ are prime) can be written as the difference of two squares.
Here is a list of all the STEP questions on Nrich.