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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 $n$?

Square Remainders

Age 16 to 18
Challenge Level

Show that every odd square leaves remainder $1$ when divided by $8$, and that every even square leaves remainder $0$ or $4$.

Deduce that a number of the form $8n+7,$ where $n$ is a positive integer, cannot be expressed as a sum of three squares.



STEP Mathematics III, Specimen paper, Q10(i). Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.