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Curvy Equation

This problem asks you to use your curve sketching knowledge to find all the solutions to an equation.

Digital Equation

Can you find a three digit number which is equal to the sum of the hundreds digit, the square of the tens digit and the cube of the units digit?

Euler's Totient Function

How many numbers are there less than $n$ which have no common factors with $n$?

Divisible Factorisations

Age 16 to 18
Challenge Level

Let $n$ be a positive integer.

  1. Factorise $n^{5}-n^{3},$ and show that it is divisible by $24$.
  2. Prove that $2^{2n}-1$ is divisible by $3$.
  3. If $n-1$ is divisible by $3$, show that $n^{3}-1$ is divisible by $9$.


STEP Mathematics I, 1996, Q3. Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.