Challenge Level

Let $k$ be an integer satisfying $0\le k \le 9\,$.

Show that $0\le 10k-k^2\le 25$.

Show also that $k(k-1)(k+1)$ is divisible by $3\,$.

Since $k$ is an integer, you could calculate the values of $10k-k^2$ for $k=0, 1, \cdots, 8, 9$. Alternatively you could consider the graph of $y=10x-x^2$.

When considering $k(k-1)(k+1)$ it might be helpful to rearrange the factors.

It might also help to consider the cases $k=3b$, $k=3b+1$ and $k=3b+2$.

*Why does this cover all possible values of $k$?*

When considering $k(k-1)(k+1)$ it might be helpful to rearrange the factors.

It might also help to consider the cases $k=3b$, $k=3b+1$ and $k=3b+2$.

For each $3$-digit number $N$, where $N\ge100$, let $S$ be the sum of the hundreds digit, the square of the tens digit and the cube of the units digit. Find the numbers $N$ such that $S=N$.

[Hint: write $N=100a+10b+c\,$ where $a\,$, $b\,$ and $c$ are the digits of $N\,$.]

Can you write down an equation connecting $a$, $b$ and $c$. You might find it helpful to have $a$ and $b$ on one side of the equation and $c$ on the other.

What values do you know that $a$, $b$, $c$ lie between? Since $N$ is a three digit number we know that $a \neq 0$.

Can you see how the first parts of the question might be related to this part?

What values do you know that $a$, $b$, $c$ lie between? Since $N$ is a three digit number we know that $a \neq 0$.

Can you see how the first parts of the question might be related to this part?

*STEP Mathematics 1, 2003, Q7. Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.*

Although number theory - the study of the natural numbers - does not typically feature in school curricula it plays a leading role in university at first year and beyond. Having a good grasp of the fundamentals of number theory is useful across all disciplines of mathematics. Moreover, problems in number theory are a great leisure past time as many require only minimal knowledge of mathematical 'content'.