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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.
This problem asks you to use your curve sketching knowledge to find all the solutions to an equation.
How many numbers are there less than $n$ which have no common factors with $n$?
Frosty the Snowman is melting. Can you use your knowledge of differential equations to find out how his volume changes as he shrinks?