Digital Equation

Age 16 to 18
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 $k$?


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?


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.


Did you know ... ?

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'.