Let the fantastic wealth of resources below teach you all about Further Number Theory.

Covers the whole of the AH Maths course

Created by an experienced maths teacher

Resources used with students in Scottish Secondary Schools

Welcome to **advancedhighermaths.co.uk**

A sound understanding of * Further Number Theory* is essential to ensure exam success.

Study at Advanced Higher Maths level will provide excellent preparation for your studies when at university. Some universities may require you to gain a pass at AH Maths to be accepted onto the course of your choice. The AH Maths course is fast paced so please do your very best to keep on top of your studies.

For students looking for extra help with the AH Maths course in 2020/21 you may wish to consider subscribing to the fantastic additional * exam focused* resources available in the Online Study Pack.

We hope you find this website useful and wish you the very best with your studies in 2020/21.

Please find below:

1. AH Maths Past Exam Worksheets by Topic

3. Further Number Theory Resources

4. MIA Text Book Questions – Further Number Theory

5. About Further Number Theory

6. AH Maths Past & Practice Papers

7. AH Maths Past Paper Questions by Topic

8. AH Maths Recommended Text Book

9. Exam Focused Online Study Pack – *For students looking for a ‘good’ Pass*

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**1. AH Maths Past Paper Exam Worksheets by Topic**

Thanks to the SQA for making these available. The worksheets by topic below are an excellent study resource since they are actual SQA past paper exam questions.

Clear, easy to follow, step-by-step worked solutions to all SQA Higher Maths Questions below are available in the Online Study Pack.

**2. Links and Exam Formulae
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AH Maths - Home Page Link | HERE |

AH Maths - Whole Course Link | HERE |

AH Maths - Exam Formula List | HERE |

Higher Maths - Exam Formulae List | HERE |

AH Maths - Complete Formula List | HERE |

SQA Full Course Content and Outline | HERE |

**3. Further Number Theory Resources**

Thanks to the author(s) for making these available.

Further Number Theory Guide | HERE |

**4. MIA Text Book Questions – Further Number Theory **

Recommended questions from the Maths In Action (2nd Edition) by Edward Mullan text book are shown below.

Worked solutions to all questions below are available in the Online Study Pack.

Subtopic _______________________________________ | Page Number _____________ | Exercise _________ | Recommended Questions ____________________________ |

Finding the Greatest Common Divisor (GCD) | Page 318 | Ex 16.3 | Q1a,c,e,g,i |

Expressing GCD in the form xa + yb = d | Page 320 | Ex 16.4 | Q1,2,3,4 |

Number Bases | Page 322 | Ex 16.5 | Q1a-d,2a-f |

Further Number Theory - SQA Questions | SQA Questions & Answers |

**5. About Further Number Theory**

For a more detailed explanation, please first read the Theory Guides above.

At AH Maths, the following is covered in the topic ‘Further Number Theory’:

- Proof by Induction – Summations only
- Euclidean Algorithm
- Linear Combinations of the GCD
- Base-n Calculations

A common style AH Maths exam question on Further Number Theory is;

Source: SQA AH Maths Paper 2015 Question 7

**Worked Solution**

**Step 1 – Start by finding the Greatest Common Divisor (GCD)**

3066p + 713q = 1 —– (1)

Euclidean Algorithm Process ….

⇒ 3066 = 4 × 713 + 214 —– (2)

⇒ 713 = 3 × 214 + 71 —– (3)

⇒ 214 = 3 × 71 + 1 —– (4)

⇒ 71 = 71 × 1 + 0 —– (5)

Hence the GCD (3066,713) = 1 … *relatively prime!*

**Step 2 – Now work backwards using equations (4), (3) & (2) above**

⇒ 214 = 3.71 + 1 —– (4)

⇒ 1 = 214 – 3.71 … re-arranging eqn (4)

⇒ 1 = 214 – 3.(713 – 3.214) … substituting 71 for eqn (3)

⇒ 1 = 214 – 3.713 + 9.214 … multiplying out the brackets

⇒ 1 = 10.214 – 3.713 … simplifying

⇒ 1 = 10.(3066 – 4.713) – 3.713 … substituting 214 for eqn (2)

⇒ 1 = 10.3066 – 40.713 – 3.713 … multiplying out the brackets

⇒ 1 = 10.3066 – 43.713 … simplifying

**Step 3 – Compare with equation (1)**

1 = 10.3066 – 43.713

1 = 3066p + 713q —– (1)

Hence p = 10, q = – 43

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**6. AH Maths Past & Practice Papers**

Thanks to the SQA for making these available.

Worked solutions to SQA Exam Questions available in the Online Study Pack.

Year ____ | Paper Type _________________ | Exam Paper ______________ | Marking Scheme _______________________________________ |

2019 | AH Specimen | Specimen | Marking Scheme |

2019 | Advanced Higher | Exam Paper | Marking Scheme |

2018 | Advanced Higher | Exam Paper | Marking Scheme |

2017 | Advanced Higher | Exam Paper | Marking Scheme |

2016 | Advanced Higher | Exam Paper | Marking Scheme |

2016 | AH Specimen | Specimen | Marking Scheme |

2016 | AH Exemplar | Exemplar | Marking Scheme |

2015 | Advanced Higher | Exam Paper | Marking Scheme |

2014 | Advanced Higher | Exam Paper | Marking Scheme |

2013 | Advanced Higher | Exam Paper | Marking Scheme |

2012 | Advanced Higher | Exam Paper | Marking Scheme |

2011 | Advanced Higher | Exam Paper | Marking Scheme |

2010 | Advanced Higher | Exam Paper | Marking Scheme |

2009 | Advanced Higher | Exam Paper | Marking Scheme |

2008 | Advanced Higher | Exam Paper | Marking Scheme |

2007 | Advanced Higher | Exam Paper | Marking Scheme |

2006 | Advanced Higher | Exam Paper | Marking Scheme |

2005 | Advanced Higher | Exam Paper | Marking Scheme |

2004 | Advanced Higher | Exam Paper | Marking Scheme |

2003 | Advanced Higher | Exam Paper | Marking Scheme |

2002 | Advanced Higher | Exam Paper | Marking Scheme |

2001 | Advanced Higher | Exam Paper | Marking Scheme |

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**7. AH Maths Past Paper Questions by Topic**

Thanks to the SQA for making these available.

Questions and answers have been split up by topic for ease of reference.

Worked solutions to all SQA questions below available in the Online Study Pack.

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**8. ****AH Maths Text Book – Maths In Action (2nd Edition) by Edward Mullan **

A fully revised course for the new Curriculum for Excellence examination that is designed to fully support the course’s new structure and unit assessment. A part of the highly regarded Maths in Action series, it provides students with a familiar, clear and carefully structured learning experience that encourages them to build confidence and understanding.

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**9. Exam focused Online Study Pack
**

Through step-by-step worked solutions to exam questions available in the Online Study Pack we cover everything you need to know about * Further Number Theory* to pass your final exam.

As well as students studying Advanced Higher Mathematics, the resources will benefit young adults studying A-Level Mathematics and undergraduates who need a little extra help. Particular benefit will be to students who have gained a ‘Conditional’ University place and are therefore required to pass in order to gain entry onto the course of their choice.

Subscribing to the Online Study Pack may therefore be one of your best ever investments. Please click Online Study Pack to view screenshots, examples and instructions on how to subscribe.

Please speak with your mum/dad and show them our fantastic website.

We hope the resources on this website prove useful and wish you the very best of success with your AH Maths course in 2020/21.

Topic Links

Free resources to dozens of AH Maths topics are available by clicking on any of the links to the right.