Modular arithmetic and divisibility testsEdexcel A-Level Further Maths: Subtopic test

1. 1
   The integers $a$ and $b$ satisfy $a\equiv4\pmod9$ and $b\equiv7\pmod9$.

   (a)

   Find the least non-negative residue of $a+b$ modulo $9$.

   [1 mark]

   • A$11$
   • B$3$
   • C$1$
   • D$2$

   (b)

   Find the least non-negative residue of $ab$ modulo $9$.

   [1 mark]

   • A$6$
   • B$1$
   • C$2$
   • D$28$

   (c)

   Find the least positive residue of $a^2-3b$ modulo $9$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The six-digit number $N=372\,416$.

   (a)

   Which one of the following statements about $N$ is true?

   [1 mark]

   • A$N$ is divisible by $4$ but not by $3$
   • B$N$ is divisible by $3$ but not by $4$
   • C$N$ is divisible by $9$
   • D$N$ is divisible by $5$

   (b)

   The final digit $6$ is replaced by a digit $d$ so that the new number is divisible by $9$. Find $d$.

   [1 mark]

   • A$0$
   • B$3$
   • C$1$
   • D$9$

   (c)

   Use a divisibility test to show that $N$ is divisible by $11$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Work modulo $11$.

   (a)

   Show that $3^5\equiv1\pmod{11}$.

   [3 marks]

   (b)

   Hence find the remainder when $3^{102}+7$ is divided by $11$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The five-digit number $N=4a56b$ has digits $4,\,a,\,5,\,6,\,b$, where $a$ and $b$ are single digits.

   (a)

   Given that $N$ is divisible by both $4$ and $9$, use divisibility tests to find all possible pairs $(a,b)$.

   [6 marks]

   (b)

   Determine which of the numbers found in part (a) is divisible by $11$. Hence explain why that number is divisible by $396$, and find its quotient when divided by $396$.

   [6 marks]

   Total for question 4: 12 marks

End of questions