plumiDivisibility and modular arithmetic: OCR MEI A‑Level Further Maths questions
OCR MEI A-Level Further Maths past-paper questions on 7.4 Divisibility and modular arithmetic, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.
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All 9 questions on Divisibility and modular arithmetic, as a list
- 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, 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, 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, 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, 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]
- 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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