Square Remainders

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.