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Groups and number theory: OCR A‑Level Further Maths questions

Every OCR 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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648 questions

  1. Q1

    June 2025 · Additional Pure Mathematics

    4 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2025 Additional Pure Mathematics, question 1: (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

  2. Q3

    June 2025 · Additional Pure Mathematics

    6 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2025 Additional Pure Mathematics, question 3: 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] +

  3. Q5

    June 2025 · Additional Pure Mathematics

    6 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2025 Additional Pure Mathematics, question 5: (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

  4. Q6

    June 2025 · Additional Pure Mathematics

    13 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2025 Additional Pure Mathematics, question 6: In this question, the set S is {2, 4, 6, 8}, and is the operation of multiplication modulo 10. # 10 (a) The group G consists of the set S, under the operation .

  5. Q1

    June 2025 · AS Additional Pure Mathematics

    4 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2025 AS Additional Pure Mathematics, question 1: 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

  6. Q4

    June 2025 · AS Additional Pure Mathematics

    13 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2025 AS Additional Pure Mathematics, question 4: The binary operation * is defined on the set A = {1, 3, 5, 7, 9} by x * y = x + y + 3 (mod 10) for all x, y ! A. (a) (i) Show that * is associative on the eleme

  7. Q7

    June 2025 · AS Additional Pure Mathematics

    9 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2025 AS Additional Pure Mathematics, question 7: A n n B n n Consider the two arithmetic sequences { 2 7 : } and { 3 1 : } . = + = + ! N ! N n n h A B for each chosen value of n. Let hcf ( , ) = n n (a) (i) Fi

  8. Q1

    June 2024 · Additional Pure Mathematics

    6 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2024 Additional Pure Mathematics, question 1: (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

  9. Q3

    June 2024 · Additional Pure Mathematics

    6 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2024 Additional Pure Mathematics, question 3: 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

  10. Q8

    June 2024 · Additional Pure Mathematics

    15 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2024 Additional Pure Mathematics, question 8: The group G is cyclic and of order 12. (a) (i) State the possible orders of all the proper subgroups of G. You must justify your [2] answers. (ii) List all the

  11. Q3

    June 2024 · AS Discrete Mathematics

    6 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2024 AS Discrete Mathematics, question 3: Heidi has a pack of cards. Each card has a single digit on one side and is blank on the other side. Each of the digits from 1 to 9 appears on exactly four cards

  12. Q1

    June 2024 · AS Additional Pure Mathematics

    2 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2024 AS Additional Pure Mathematics, question 1: 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]

  13. Q5

    June 2024 · AS Additional Pure Mathematics

    14 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2024 AS Additional Pure Mathematics, question 5: The set S consists of all 2◊2 matrices having determinant 1 or -1. For instance, the matrices J N J N 3 3 J N 1 1 1 0 K O K O 2 2 2 2 KK OO are elements of S. I

  14. Q6

    June 2024 · AS Additional Pure Mathematics

    9 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2024 AS Additional Pure Mathematics, question 6: 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

  15. Q5

    June 2023 · Additional Pure Mathematics

    10 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2023 Additional Pure Mathematics, question 5: = " , 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

  16. Q8

    June 2023 · Additional Pure Mathematics

    9 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2023 Additional Pure Mathematics, question 8: 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

  17. Q9

    June 2023 · Additional Pure Mathematics

    11 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2023 Additional Pure Mathematics, question 9: The set C consists of the set of all complex numbers excluding 1 and - . The operation 5 is 1 a b + defined on the elements of C by a b where ,a b C . 5 = ! ab

  18. Q1

    June 2023 · AS Additional Pure Mathematics

    3 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2023 AS Additional Pure Mathematics, question 1: (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

  19. Q2

    June 2023 · AS Additional Pure Mathematics

    4 marksFP.7 Groups and number theoryFP.9 Further sequences and series
    The whole paper ↗
    June 2023 AS Additional Pure Mathematics, question 2: " , 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

  20. Q5

    June 2023 · AS Additional Pure Mathematics

    11 marksFP.7 Groups and number theory
    The whole paper ↗
    June 2023 AS Additional Pure Mathematics, question 5: (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
All 48 questions on Groups and number theory, as a list
  1. 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
  2. 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] +
  3. 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
  4. June 2025, Additional Pure Mathematics, Q6 (13 marks): In this question, the set S is {2, 4, 6, 8}, and is the operation of multiplication modulo 10. # 10 (a) The group G consists of the set S, under the operation .
  5. 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
  6. June 2025, AS Additional Pure Mathematics, Q4 (13 marks): The binary operation * is defined on the set A = {1, 3, 5, 7, 9} by x * y = x + y + 3 (mod 10) for all x, y ! A. (a) (i) Show that * is associative on the eleme
  7. June 2025, AS Additional Pure Mathematics, Q7 (9 marks): A n n B n n Consider the two arithmetic sequences { 2 7 : } and { 3 1 : } . = + = + ! N ! N n n h A B for each chosen value of n. Let hcf ( , ) = n n (a) (i) Fi
  8. 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
  9. 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
  10. June 2024, Additional Pure Mathematics, Q8 (15 marks): The group G is cyclic and of order 12. (a) (i) State the possible orders of all the proper subgroups of G. You must justify your [2] answers. (ii) List all the
  11. June 2024, AS Discrete Mathematics, Q3 (6 marks): Heidi has a pack of cards. Each card has a single digit on one side and is blank on the other side. Each of the digits from 1 to 9 appears on exactly four cards
  12. 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]
  13. June 2024, AS Additional Pure Mathematics, Q5 (14 marks): The set S consists of all 2◊2 matrices having determinant 1 or -1. For instance, the matrices J N J N 3 3 J N 1 1 1 0 K O K O 2 2 2 2 KK OO are elements of S. I
  14. 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
  15. 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
  16. 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
  17. June 2023, Additional Pure Mathematics, Q9 (11 marks): The set C consists of the set of all complex numbers excluding 1 and - . The operation 5 is 1 a b + defined on the elements of C by a b where ,a b C . 5 = ! ab
  18. 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
  19. 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
  20. 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
  21. June 2023, AS Additional Pure Mathematics, Q7 (14 marks): The group G, of order 12, consists of the set {1, 2, 4, 5, 8, 10, 13, 16, 17, 19, 20, x} under the operation of multiplication modulo 21. The identity of G is t
  22. June 2022, Additional Pure Mathematics, Q2 (5 marks): Consider the integers a and b, where, for each integer n, a 7 n + and b 4 8 n + . 5 = = Let h hcf ( , ) a b . = (a) Determine all possible values of h. [3] (b)
  23. 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
  24. June 2022, Additional Pure Mathematics, Q8 (10 marks): (a) Explain why all groups of even order must contain at least one self-inverse element (that is, an element of order 2). [2] (b) Prove that any group, in which
  25. 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
  26. 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
  27. June 2022, AS Additional Pure Mathematics, Q7 (13 marks): The diagram below shows an equilateral triangle ABC. The three lines of reflection symmetry of ABC (the lines a, b and c) are shown as broken lines. The point o
  28. 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]
  29. November 2021, Additional Pure Mathematics, Q2 (5 marks): The following Cayley table is for G, a group of order 6. The identity element is e and the group is generated by the elements a and b. a2 2 G e a b ab a b a2 2
  30. 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
  31. November 2021, Additional Pure Mathematics, Q6 (10 marks): The binary operation Z is defined on the set ℂ of complex numbers by ( a i b ) ( c i d ) ac i ( b ad ) + Z + = + + where a, b, c and d are real numbers. (a) Is
  32. 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
  33. 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)
  34. November 2021, AS Additional Pure Mathematics, Q6 (11 marks): The set S consists of the following four complex numbers. 3 i 3 i 1 i 3 1 i 3 + - - - - + ^ h 1 For z1, z S , the binary operation & is defined by z z 1 i 3 z z
  35. 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
  36. November 2020, Additional Pure Mathematics, Q1 (5 marks): The following Cayley table is for a set {a, b, c, d} under a suitable binary operation. a b c d a b a b c c d d a (a) Given that the Latin square property holds
  37. November 2020, Additional Pure Mathematics, Q6 (10 marks): The group G consists of the set {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36} under , the operation # 39 of multiplication modulo 39. (a) List the possible orde
  38. 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 -
  39. 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] /
  40. 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
  41. November 2020, AS Additional Pure Mathematics, Q4 (12 marks): (a) For the set S { , , , , 2 4 6 8 10 12 , }, under the operation #14 of multiplication modulo 14, = complete the Cayley table given in the Printed Answer Book
  42. 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] /
  43. June 2019, Additional Pure Mathematics, Q5 (11 marks): The group G consists of a set S together with , the operation of multiplication modulo 80. It is # 80 given that S is the smallest set which contains the elemen
  44. 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
  45. June 2019, AS Discrete Mathematics, Q1 (5 marks): Alfie has a set of 15 cards numbered consecutively from 1 to 15. He chooses two of the cards. (a) How many different sets of two cards are possible? [2] Alfie p
  46. 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]
  47. 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]
  48. June 2019, AS Additional Pure Mathematics, Q7 (12 marks): You are given the set S = {1, 5, 7, 11, 13, 17} together with , the operation of multiplication # 18 modulo 18. (a) Complete the Cayley table for (S, ) given in

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