### Introduction

This STEP support module includes some work with summations, and an introduction to Modular Arithmetic. Previous assignments can be found here, but you can do this one without having done the others first.

STEP questions are difficult, they are supposed to be and you should expect to get stuck. However, as you tackle more and more STEP questions you will develop a range of problem solving skills (and spend less time "being stuck").

### About this assignment

The assignment is published as a pdf file below. Each STEP Support assignment module starts with a warm-up exercise, followed by preparatory work leading to a STEP question. Finally, there is a warm-down exercise.

The **warm up** for this assignment introduces Modular Arithmetic and shows how you can use this to find divisibility tests for dividing by 3 and 11.

The **main STEP question (2003 STEP 1 Question 1)** derives the results for the sum of $n$ squares and $n$ cubes.

The **final question** is about using modulo arithmetic to prove that expressions are divisible by certain numbers (these sorts of questions are more usually presented as exercises in proof by induction, but using modulo arithmetic can often be a simpler approach).

There are lots of articles, discussions and problems involving modular arithmetic out there. For starters, try this Maths in a minute article, this Nrich page and this page.

### Hints, support and self evaluation

The **“Hints and partial solutions for Assignment 17”** file gives suggestions on how you can tackle the questions, and some common pitfalls to avoid, as well as some partial solutions and answers.

Here is a Worked Video Solution to the STEP question from this assignment.