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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 $3\,$.
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 $N\,$.]
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.