1. 1
   A gardener is planting bulbs in straight rows. She has 60 tulip bulbs and 84 daffodil bulbs. She wants to arrange all the bulbs into rows so that every row has the same number of bulbs, and each row contains only one type of bulb.

   (a)

   What is the highest common factor (HCF) of 60 and 84?

   [1 mark]

   • A$6$
   • B$12$
   • C$24$
   • D$420$

   (b)

   Written as a product of its prime factors, $60 = 2^2 \times 3 \times 5$. Which of the following is the correct prime factorisation of 84?

   [1 mark]

   • A$2^2 \times 3 \times 7$
   • B$2 \times 3 \times 7^2$
   • C$2^2 \times 21$
   • D$2^3 \times 3 \times 7$

   (c)

   The gardener wants to know the smallest number of tulip bulbs she would need to buy in total if she instead planted tulips in groups of both 60 and 84 (i.e. the lowest common multiple). What is the LCM of 60 and 84?

   [1 mark]

   • A$168$
   • B$5040$
   • C$12$
   • D$420$

   Total for question 1: 3 marks

2. 2
   Two coach companies run services from the same station. Coach A departs every 90 minutes and Coach B departs every 126 minutes. Both coaches depart together at 08:00.

   (a)

   Write 90 as a product of its prime factors. Give your answer in index notation.

   [2 marks]

   (b)

   Given that $126 = 2 \times 3^2 \times 7$, find the lowest common multiple of 90 and 126. Use your answer to state the next time, after 08:00, that both coaches will depart together.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A factory produces metal rods in two batch sizes: batch X contains 132 rods and batch Y contains 198 rods. A quality inspector wants to divide each batch into the largest possible equal groups of rods, with no rods left over, so that every group (from either batch) contains the same number of rods.

   (a)

   Show that the highest common factor of 132 and 198 is 66.

   [3 marks]

   (b)

   Use your answer to part (a) to find the total number of equal-sized groups the inspector can make from both batches combined, if each group contains 66 rods.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   Three warning lights at a warehouse flash at regular intervals. Light P flashes every $2^3 \times 3$ seconds, light Q flashes every $2 \times 3^2$ seconds, and light R flashes every $2^2 \times 3 \times 5$ seconds. All three lights flash together at the same instant.

   (a)

   Find, using prime factorisation, the number of seconds until all three lights next flash together at the same instant.

   [4 marks]

   (b)

   Using the prime factorisations given in the context, find the highest common factor of the three flashing intervals ($2^3 \times 3$, $2 \times 3^2$ and $2^2 \times 3 \times 5$ seconds).

   [4 marks]

   (c)

   A whole number $n$ is such that $n = 2^a \times 3^b$, where $a$ and $b$ are positive integers. The HCF of $n$ and $360$ is $2^2 \times 3^2$, and the LCM of $n$ and $360$ is $2^3 \times 3^2 \times 5$. Given that $360 = 2^3 \times 3^2 \times 5$, find the value of $n$, showing your full reasoning.

   [5 marks]

   Total for question 4: 13 marks

End of questions