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STEP Support Programme

STEP Support - Assignment 17


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).

Assignment PDF: 
Hints and Partial Solutions: 

Useful Links

Underground Mathematics: Selected worked STEP questions

STEP Question database

University of Cambridge Mathematics Faculty: What do we look for?

University of Cambridge Mathematics Faculty: Information about STEP

University of Cambridge Admissions Office: Undergraduate course information for Mathematics

Stephen Siklos' "Advanced Problems in Mathematics" book (external link)

MEI: Worked solutions to STEP questions (external link)

OCR: Exam board information about STEP (external link)

AMSP (Advanced Maths Support programme): Support for University Admission Tests (external link)