plumiGroups and number theory: OCR MEI A‑Level Further Maths questions
Every OCR MEI A-Level Further Maths exam question on FP.7 Groups and number theory, 2018–2025, with its mark scheme and examiner’s report.
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![June 2025 Extra Pure, question 1: A binary operation, ○, is defined on real numbers a and b by a ○ b 3 ab . = (a) Prove that ○ is associative over the real numbers R. [2] (b) Determine the value](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y43501-jun25-01-qp.webp?v=3)
![June 2025 Further Pure with Technology, question 2: 2 (a) (i) Write down 7 ( mod 1000 ) . [1] Fermat’s little theorem states that if p is a prime and x is an integer which is co-prime to p, 1 / p then x - 1 ( mod](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y43601-jun25-02-qp.webp?v=3)
![June 2024 Further Pure with Technology, question 2: Wilson’s theorem states that a positive integer n 2 is prime if and only if 1 ( n 1 )! n 1 ( mod n ) . - / - (a) (i) Calculate 136 !( mod 137 . ) [1] (ii) Hence](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y43601-jun24-02-qp.webp?v=3)
![June 2023 Further Pure with Technology, question 3: Wilson’s theorem states that an integer p 2 1 is prime if and only if ( p 1 )! 1 ( mod p ) . - - / (a) Use Wilson’s theorem to show that 17 ! 1 ( mod 19 ) . [2]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y43601-jun23-03-qp.webp?v=3)
![November 2021 Extra Pure, question 2: G is a group of order 8. (a) Explain why there is no subgroup of G of order 6. [1] You are now given that G is a cyclic group with the following features: • e i](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y43501-nov21-02-qp.webp?v=3)
![November 2021 Further Pure with Technology, question 3: (a) (i) Create a program which returns the highest common factor of positive integers m and n. Write out your program in full in the Printed Answer Booklet. [3]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y43601-nov21-03-qp.webp?v=3)
![June 2019 Extra Pure, question 6: ! 2 6 (a) Given that 7 is an irrational number, prove that a 7 b 0 for all ,a b where a and b - ! Q are not both 0. [2] (b) A set G is defined by G { a b 7 : ,a](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y43501-jun19-06-qp.webp?v=3)
![June 2019 Further Pure with Technology, question 2: 2 2 (a) Prove that if x and y are integers which satisfy x 2 y = , then x is odd and y is even. 1 [3] - (b) Create a program to find, for a fixed positive integ](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y43601-jun19-02-qp.webp?v=3)
All 18 questions on Groups and number theory, as a list
- June 2025, Extra Pure, Q1 (7 marks): A binary operation, ○, is defined on real numbers a and b by a ○ b 3 ab . = (a) Prove that ○ is associative over the real numbers R. [2] (b) Determine the value
- June 2025, Extra Pure, Q3 (8 marks): M is the group ({0, 1, 2, 3, 4, 5, 6, 7, 8}, + ) where + denotes the binary operation of addition 9 9 modulo 9. (a) Write down the identity element of M. Justif
- June 2025, Further Pure with Technology, Q2 (21 marks): 2 (a) (i) Write down 7 ( mod 1000 ) . [1] Fermat’s little theorem states that if p is a prime and x is an integer which is co-prime to p, 1 / p then x - 1 ( mod
- June 2024, Extra Pure, Q3 (12 marks): Fig. 3.1 shows an equilateral triangle, with vertices A, B and C, and the three axes of symmetry of the triangle, S S , b and Sc. The axes of symmetry are fixed
- June 2024, Extra Pure, Q5 (4 marks): In this question you may assume that if p and q are distinct prime numbers and p a q b where = , , then 0 and 0 . Z a b ! a = b = a (a) Prove that it is not pos
- June 2024, Further Pure with Technology, Q2 (23 marks): Wilson’s theorem states that a positive integer n 2 is prime if and only if 1 ( n 1 )! n 1 ( mod n ) . - / - (a) (i) Calculate 136 !( mod 137 . ) [1] (ii) Hence
- June 2023, Extra Pure, Q4 (15 marks): The set G is given by G = {M: M is a real 2 # matrix and det M 2 = }. 1 (a) Show that G forms a group under matrix multiplication, #. You may assume that matrix
- June 2023, Further Pure with Technology, Q2 (11 marks): Throughout this question (a, b, c) is a Pythagorean triple with the positive integers a, b, c ordered such that a G b G c. 2 = (a) Show that a b + if and only i
- June 2023, Further Pure with Technology, Q3 (10 marks): Wilson’s theorem states that an integer p 2 1 is prime if and only if ( p 1 )! 1 ( mod p ) . - - / (a) Use Wilson’s theorem to show that 17 ! 1 ( mod 19 ) . [2]
- June 2022, Extra Pure, Q4 (16 marks): A binary operation, , is defined on a set of numbers, A, in the following way. a b = k a k b k , for a, b ! A, - + 1 2 3 where k , k and k 3 are constants (whic
- June 2022, Further Pure with Technology, Q2 (20 marks): (a) In this part of the question n is an integer greater than 1. An integer q, where 0 1 q 1 n is said to be a quadratic residue modulo n if there exists an 2 /
- November 2021, Extra Pure, Q2 (7 marks): G is a group of order 8. (a) Explain why there is no subgroup of G of order 6. [1] You are now given that G is a cyclic group with the following features: • e i
- November 2021, Extra Pure, Q6 (8 marks): You are given that q with q H and that 1 ! Z 1 1 1 S . = + + + f ( q 1 ) ( q 1 )( q 2 ) ( q 1 )( q 2 )( q 3 ) + + + + + + 1 (a) By considering a suitable geomet
- November 2021, Further Pure with Technology, Q3 (20 marks): (a) (i) Create a program which returns the highest common factor of positive integers m and n. Write out your program in full in the Printed Answer Booklet. [3]
- November 2020, Extra Pure, Q4 (13 marks): (a) In each of the following cases, a set G and a binary operation are given. The operation may be assumed to be associative on G. Determine which, if any, of t
- June 2019, Extra Pure, Q4 (8 marks): T is the set {1, 2, 3, 4}. A binary operation • is defined on T such that a • a = 2 for all a ! T. It is given that (T, •) is a group. (a) Deduce the identity e
- June 2019, Extra Pure, Q6 (13 marks): ! 2 6 (a) Given that 7 is an irrational number, prove that a 7 b 0 for all ,a b where a and b - ! Q are not both 0. [2] (b) A set G is defined by G { a b 7 : ,a
- June 2019, Further Pure with Technology, Q2 (20 marks): 2 2 (a) Prove that if x and y are integers which satisfy x 2 y = , then x is odd and y is even. 1 [3] - (b) Create a program to find, for a fixed positive integ
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