plumiGroups and number theory: Edexcel A‑Level Further Maths questions
Every Edexcel 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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All 35 questions on Groups and number theory, as a list
- June 2025, Further Pure 2, Q1 (7 marks): The set S = {1, 3, 5, 9, 11, 13} forms the group G, under the operation multiplication modulo 14 (a) Complete the Cayley table below for the group G ×14 1 3 5 9
- June 2025, Further Pure 2, Q5 (10 marks): (i) A password is to be created from 5 characters using the following criteria • one capital letter • 3 distinct single digits • one of the four symbols @ £ $ &
- June 2025, Further Pure 2, Q7 (6 marks): (i) Explain why the congruence equation 21x ≡ 11(mod 36) has no solutions. (2) (ii) Use modular arithmetic to solve the congruence equation 36y ≡ 192(mod 40) (4
- June 2025, AS Further Pure 2, Q1 (8 marks): In this question you must show all stages of your working. Solutions based entirely on calculator technology are not acceptable. (i) (a) Use the Euclidean algor
- June 2025, AS Further Pure 2, Q2 (8 marks): (i) Using a suitable algorithm and without performing any division, determine whether 13 306 617 is divisible by 9 (2) (ii) The group G = {1, 3, 7, 9, 11, 13, 1
- June 2024, Further Pure 2, Q1 (4 marks): In this question you must show detailed reasoning. Use Fermat’s Little Theorem to determine the least positive residue of 21 80 (mod 23) (4)
- June 2024, Further Pure 2, Q3 (11 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. (a) Use the Euclidean Algorithm to det
- June 2024, Further Pure 2, Q7 (10 marks): The set of matrices G = {I, A, B, C, D, E} where I A B C D 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 1 E 0 1 1 1 with the operation Ä2
- June 2024, AS Further Pure 2, Q1 (9 marks): (i) The table below is a Cayley table for the group G with operation ° ° a b c d e f a d c b a f e b e f a b c d c f e d c b a d a b c d e f e b a f e d c f c d
- June 2024, AS Further Pure 2, Q2 (6 marks): Tiles are sold in boxes with 21 tiles in each box. The tiles are laid out in x rows of 5 tiles and y rows of 6 tiles. All the tiles from a box are used before t
- June 2023, Further Pure 2, Q4 (9 marks): (a) Use the Euclidean algorithm to show that the highest common factor of 168 and 66 is 6 (2) (b) Use back substitution to determine integers a and b such that
- June 2023, Further Pure 2, Q5 (8 marks): (i) A security code is made up of 4 numerical digits followed by 3 distinct uppercase letters. Given that the digits must be from the set {1, 2, 3, 4, 5} and th
- June 2023, Further Pure 2, Q7 (6 marks): The set G 3 2 with the operation of x y x y xy 3 1 2 forms a group. (a) Determine the identity element of this group. (2) (b) Determine the inverse of a gener
- June 2023, AS Further Pure 2, Q1 (8 marks): The operation * is defined on the set G = {0, 1, 2, 3} by x * y ≡ x + y – 2xy (mod 4) (a) Complete the Cayley table below. (2) * 0 1 2 3 0 1 2 3 (b) Show that G
- June 2023, AS Further Pure 2, Q5 (8 marks): (i) Making your reasoning clear and using modulo arithmetic, show that 214 6 is divisible by 8 (3) (ii) The following 7‑digit number has four unknown digits a 5
- June 2022, Further Pure 2, Q1 (6 marks): The group S4 is the set of all possible permutations that can be performed on the four numbers 1, 2, 3 and 4, under the operation of composition. For the group
- June 2022, Further Pure 2, Q4 (7 marks): (a) Use the Euclidean algorithm to show that 124 and 17 are relatively prime (coprime). (2) (b) Hence solve the equation 124x + 17y = 10 (3) (c) Solve the congr
- June 2022, Further Pure 2, Q7 (8 marks): (i) The polynomial F(x) is a quartic such that F(x) = px 4 + qx 3 + 2x 2 + rx + s where p, q, r and s are distinct constants. Determine the number of possible q
- June 2022, AS Further Pure 2, Q3 (9 marks): (i) Let G be a group of order 5 291 848 Without performing any division, use proof by contradiction to show that G cannot have a subgroup of order 11 (3) (ii) (
- June 2022, AS Further Pure 2, Q4 (11 marks): In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. (i) (a) Use the Euclidean algorithm to
- November 2021, Further Pure 2, Q1 (4 marks): In this question you must show detailed reasoning. Without performing any division, explain why n = 20 210 520 is divisible by 66 (4)
- November 2021, Further Pure 2, Q2 (8 marks): A binary operation on the set of non‑negative integers, + 0, is defined by m n = ½m – n½ m, n ∈ + 0 (a) Explain why + 0 is closed under the operation
- November 2021, Further Pure 2, Q3 (8 marks): (a) Use the Euclidean Algorithm to find integers a and b such that 125a + 87b = 1 (5) (b) Hence write down a multiplicative inverse of 87 modulo 125 (1) (c) Sol
- November 2021, Further Pure 2, Q4 (7 marks): Let G be a group of order 46 46 + 47 47 Using Fermat’s Little Theorem and explaining your reasoning, determine which of the following are possible orders for a
- November 2020, Further Pure 2, Q1 (6 marks): A small sports club has 12 adult members and 14 junior members. The club needs to enter a team of 8 players for a particular competition. Determine the number o
- November 2020, Further Pure 2, Q6 (10 marks): 1 2 3 4 5 6 Figure 3 Figure 3 shows a plane shape made up of a regular hexagon with an equilateral triangle joined to each edge and with alternate equilateral t
- November 2020, Further Pure 2, Q8 (12 marks): The four digit number n = abcd satisfies the following properties: (1) n ≡ 3 (mod 7) (2) n is divisible by 9 (3) the first two digits have the same sum as the l
- November 2020, AS Further Pure 2, Q1 (8 marks): The set G = {1, 3, 7, 9, 11, 13, 17, 19} under the binary operation of multiplication modulo 20 forms a group. (a) Find the inverse of each element of G. (3) (b
- November 2020, AS Further Pure 2, Q2 (6 marks): The highest common factor of 963 and 657 is c. (a) Use the Euclidean algorithm to find the value of c. (3) (b) Hence find integers a and b such that 963a + 657b
- June 2019, Further Pure 2, Q4 (12 marks): (i) Use Fermat’s Little Theorem to find the least positive residue of 6PRGXOR (5) (ii) Seven students, Alan, Brenda, Charles, Devindra, Enid, Felix and G
- June 2019, Further Pure 2, Q6 (12 marks): (i) A binary operation * is defined on positive real numbers by a * b = a + b + ab Prove that the operation * is associative. (4) (ii) The set G ^
- June 2019, AS Further Pure 2, Q2 (7 marks): (i) Determine all the possible integers a, where a ! 3, such that 15 3 { mod a (2) (ii) Show that if p is prime, x is an integer and x p 2 1 { mod then either x
- June 2019, AS Further Pure 2, Q4 (7 marks): The set e p q r s , , , , { } forms a group, A, under the operation * Given that e is the identity element and that p*p = s s*s = r p*p*p = q (a) show that (i)
- June 2018, AS Further Pure 2, Q1 (5 marks): (i) Using a suitable algorithm and without performing any division, determine whether 23 738 is divisible by 11 (2) (ii) Use the Euclidean algorithm to find the
- June 2018, AS Further Pure 2, Q2 (10 marks): A B C O y z x Figure 1 Figure 1 shows an equilateral triangle ABC. The lines x, y and z and their point of intersection, O, are fixed in the plane. The triangle
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