Challenge Level

This problem explores some equations involving integers and number theory. It also explores how number theory facts can be used to find all integer solutions to a problem. There are several ways of approaching this problem, including checking manually all the possibilities.

STEP papers are examinations taken at the end of Year 13 and are used by Cambridge, Warwick and a number of other universities in the UK as part of their offer (or part of a range of offers) to Maths applicants. The questions are based on A Level Maths and Further Maths, but differ in style to A level questions. This question can be attempted with no A level knowledge at all and so is a good introduction to university entrance tests for Year 12 (and Year 11) students.

Students could be asked to find more than one way of showing that $0 \le 10k-k^2 \le 25$. When considering $k(k-1)(k+1)$ they could consider a few values and then try to show that it is always divisible by 3 (though the question only asks you to show it for $0\le k \le 9$).

For the second part of the question, students should start by trying to find an equation involving $a$, $b$ and $c$. They should gather up terms and then see if they can connect their equation to the first part of the question. It might be easier to isolate the $c$ terms on one side of the equation.

- If $k$ is an integer between 0 and 9, what possible values can $10k-k^2$ take? What are the minimum and maximum values?
- What do you know about the factors $k$, $k-1$ and $k+1$? What does this mean about the product of these three factors?
- What is $S$ in terms of $a$, $b$ and $c$? Make sure you read the question carefully!
- Can you use $S=N$ to write an equation involving $a$, $b$ and $c$?
- What values can $a$, $b$, $c$ take? What does the fact that $N$ is a 3-digit number tell us about $a$?
- Can you link the first parts of the question to the last part?
- Can you put bounds on the values of the parts of the equation involving $a$, $b$, and $c$?
- Can you rule out any values of $a$, $b$ or $c$?

Students could try the problem Rational Integer where they are encourage to work through possible integer values systematically. They might want to restrict themselves to positive integers to fit in with this problem more closely. Students could also consider the first question from Common Divisor.

Encourage students to consider the key questions above. A systematic approach, trying out values, is not cheating!

Students who have enjoyed this problem might like to explore the assignments from the STEP Support Programme.

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*