Divisibility and modular arithmetic: OCR A‑Level Further Maths questions

OCR 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.

All 28 questions on Divisibility and modular arithmetic, as a list

June 2025, Additional Pure Mathematics, Q1 (4 marks): (a) A student claims that the order of 5 modulo 37 is 5. Without calculation, explain why this student’s claim must be incorrect. [1] (b) Show that the order of
June 2025, Additional Pure Mathematics, Q3 (6 marks): Let N 10 a + and M b a 3 b , where a and b are positive integers and 0 b 9 . = = - G G (a) By considering N 3 M , prove that 31 | N if and only if 31 | M. [4] +
June 2025, Additional Pure Mathematics, Q5 (6 marks): (a) Determine the solution of the linear congruence x 3 7 (mod 13). Give your answer in the / form x a (mod 13), where 0 a 13 . [2] 1 / G (b) Hence or otherwise
June 2025, AS Additional Pure Mathematics, Q1 (4 marks): Use standard divisibility tests for small numbers to show that 2 573 208 is divisible by 792. [4] J N J N J N 1 2 11
June 2024, Additional Pure Mathematics, Q1 (6 marks): (a) The number N has the base -10 form N abba abba … abba, consisting of blocks of four = digits, as shown, where a and b are integers such that 1 a 10 and 0 b
June 2024, Additional Pure Mathematics, Q3 (6 marks): Determine all integers x for which x / (mod 7) and x 1 22 (mod 37) and x 7 (mod 67). / / Give your answer in the form x qn + for integers n, q, r with q r 0 and
June 2024, AS Additional Pure Mathematics, Q1 (2 marks): In this question you must show detailed reasoning. The number N is written as 28A3B in base-12 form. Express N in decimal (base-10) form. [2]
June 2024, AS Additional Pure Mathematics, Q6 (9 marks): n n 6 For positive integers n, let ( ) f n 1 2 4 . = + + (a) (i) Given that n is a multiple of 3, but not of 9, use the division algorithm to write down the two
June 2023, Additional Pure Mathematics, Q5 (10 marks): = " , under 5 (a) The group G consists of the set S 1 9 17 25 , , , , the operation of multiplication # 32 modulo 32. (i) Complete the Cayley table for G given
June 2023, Additional Pure Mathematics, Q8 (9 marks): Let ( ) f n denote the base-n number 2121n where n H . 3 (a) (i) For each n H , show that ( ) 3 f n can be written as the product of two positive integers great
June 2023, AS Additional Pure Mathematics, Q1 (3 marks): (a) Express 205 in the form q 7 + for positive integers q and r, with r 0 r 1 7 . [1] G ^ h, use the result of part (a) to justify that | (b) Given that | 7 205
June 2023, AS Additional Pure Mathematics, Q2 (4 marks): " , are given by the formula 2 For all positive integers n, the terms of the sequence un 2 u 3 n 3 n 7 ( mod 10 ). = + + n (a) Show that u u for all positive in
June 2023, AS Additional Pure Mathematics, Q5 (11 marks): (a) Express as a decimal (base-10) number the base-23 number 711923. [2] (b) Solve the linear congruence 7 n 11 9 ( mod 23 ) . [3] + / (c) Let N 10 a + and M b
June 2022, Additional Pure Mathematics, Q4 (9 marks): Let N be the number 15 824 578. (a) (i) Use a standard divisibility test to show that N is a multiple of 11. [2] (ii) A student uses the following test for divi
June 2022, AS Additional Pure Mathematics, Q4 (8 marks): Let N 10 a + and M b a 3 b , where a and b are integers such that a H and 1 0 b 9 . = = + G G (a) Prove that 29 | N if and only if 29 | M. [5] (b) Use an iterat
June 2022, AS Additional Pure Mathematics, Q6 (14 marks): The sequence { un is such that u } 7 , u 37 , u 337 , u 3337 , … . 1 = 2 = 3 = 4 = (a) Write down a first‑order recurrence system for { un . } [2] 1 n (b) By so
November 2021, Additional Pure Mathematics, Q1 (3 marks): In this question you must show detailed reasoning. Express the number 4172310 in hexadecimal (base 16). [3]
November 2021, Additional Pure Mathematics, Q4 (6 marks): Solve the simultaneous linear congruences x / (mod 3), x 1 / (mod 11), x 5 2 / (mod 17). [6] 5
November 2021, Additional Pure Mathematics, Q8 (12 marks): (a) Solve the second-order recurrence system H 5 H 4 H with H 0 = , H 3 7 for = - 1 = n n n 2 1 + + n 0 . [5] H (b) (i) Write down the quadratic residues modulo
November 2021, AS Additional Pure Mathematics, Q4 (6 marks): (a) Let a = 1071 and b = 67. (i) Find the unique integers q and r such that a bq + , where q 2 0 and 0 G r 1 b. r [1] = (ii) Hence express the answer to (a)(i)
November 2021, AS Additional Pure Mathematics, Q7 (10 marks): + . 4 n 3 7 (a) Let ( ) f n 2 + 3 3 n 1 = + Use arithmetic modulo 11 to prove that ( ) f n 0 (mod 11) for all integers n H 0. [4] / (b) Use the standard test fo
November 2020, Additional Pure Mathematics, Q7 (10 marks): Throughout this question, n is a positive integer. 5 / (a) Explain why n n (mod 5). [1] 5 / 5 / (b) By proving that n n (mod 2), show that n n (mod 10). [3] 5 -
November 2020, AS Additional Pure Mathematics, Q1 (4 marks): (a) Evaluate 13 19 modulo 31. [1] # (b) Solve the linear congruence 13 x 9 (mod 31). [3] /
November 2020, AS Additional Pure Mathematics, Q3 (6 marks): In this question, N is the number 26 132 652. (a) Without dividing N by 13, explain why 13 is a factor of N. [1] (b) Use standard divisibility tests to show tha
June 2019, Additional Pure Mathematics, Q3 (4 marks): (a) Solve x 7 6 (mod 19). [2] / (b) Show that the following simultaneous linear congruences have no solution. x / (mod 4), x 3 4 (mod 6). [2] /
June 2019, Additional Pure Mathematics, Q8 (11 marks): In this question you must show detailed reasoning. 2 / p (a) Prove that ( 2 p 2 ) 1 (mod p), where p is an odd prime. [4] - - - 34 15 (b) Find two odd prime fac
June 2019, AS Additional Pure Mathematics, Q1 (3 marks): In decimal (base 10) form, the number N is 15 260. (a) Express N in binary (base 2) form. [1] (b) Using the binary form of N, show that N is divisible by 7. [2]
June 2019, AS Additional Pure Mathematics, Q6 (8 marks): (a) Determine all values of x for which 16 x / (mod 101). 5 [4] (b) Solve (i) 95 x 6 (mod 101), [2] / (ii) 95 x / (mod 101). 5 [2]

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