Fermat's little theorem and congruence equationsEdexcel A-Level Further Maths: Subtopic test

1. 1
   Work modulo $7$, which is prime.

   (a)

   Use Fermat's little theorem to find the least non-negative residue of $4^{20}$ modulo $7$.

   [1 mark]

   • A$1$
   • B$2$
   • C$4$
   • D$6$

   (b)

   Find the least non-negative residue of $3^{100}$ modulo $7$.

   [1 mark]

   • A$1$
   • B$3$
   • C$4$
   • D$2$

   (c)

   Find the remainder when $12^{50}$ is divided by $7$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Linear congruences of the form $ax\equiv b\pmod n$ are to be solved for integers $x$.

   (a)

   Which one of the following congruences has no solutions?

   [1 mark]

   • A$3x\equiv4\pmod7$
   • B$4x\equiv6\pmod{10}$
   • C$5x\equiv2\pmod{12}$
   • D$6x\equiv4\pmod9$

   (b)

   How many solutions, modulo $15$, does $6x\equiv9\pmod{15}$ have?

   [1 mark]

   • A$3$
   • B$1$
   • C$6$
   • D$0$

   (c)

   Solve $7x\equiv3\pmod{11}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the numbers $17$ and $40$.

   (a)

   Use the Euclidean algorithm to show that $17$ and $40$ are coprime.

   [3 marks]

   (b)

   Hence find the multiplicative inverse of $17$ modulo $40$, and solve $17x\equiv9\pmod{40}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Questions on powers modulo a prime and on linear congruences.

   (a)

   Use Fermat's little theorem with the prime $13$ to show that $3^{200}+2^{100}+1$ is divisible by $13$.

   [6 marks]

   (b)

   Determine whether $15x\equiv9\pmod{21}$ has solutions. If it does, find all solutions with $0\leq x<21$.

   [6 marks]

   Total for question 4: 12 marks

End of questions