Division theorem, Euclidean algorithm and Bezout's identityEdexcel A-Level Further Maths: Subtopic test

1. 1
   Consider the integers $252$ and $198$.

   (a)

   Write $252=198q+r$ with $0\leq r<198$. Find $r$.

   [1 mark]

   • A$54$
   • B$1$
   • C$18$
   • D$36$

   (b)

   Use the Euclidean algorithm to find the highest common factor of $252$ and $198$.

   [1 mark]

   • A$6$
   • B$9$
   • C$36$
   • D$18$

   (c)

   Find the lowest common multiple of $252$ and $198$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A florist has 391 roses and 221 lilies. She makes identical bouquets, using every flower, with each bouquet containing the same number of roses and the same number of lilies.

   (a)

   Find the largest number of bouquets she can make.

   [1 mark]

   • A$51$
   • B$170$
   • C$17$
   • D$13$

   (b)

   How many roses are in each bouquet when she makes the largest possible number of bouquets?

   [1 mark]

   • A$13$
   • B$23$
   • C$17$
   • D$170$

   (c)

   Explain why $\text{hcf}(391,221)=\text{hcf}(221,170)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $a=240$ and $b=46$.

   (a)

   Use the Euclidean algorithm to find the highest common factor of $a$ and $b$.

   [3 marks]

   (b)

   Hence find integers $x$ and $y$ such that $240x+46y=2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A rectangular patio measures 1547 cm by 504 cm. It is to be covered exactly with identical square tiles, with no tile cut.

   (a)

   Use the Euclidean algorithm to find the side length of the largest square tile that can be used. Find the number of tiles this needs.

   [6 marks]

   (b)

   (i) Find integers $x$ and $y$ such that $1547x+504y=7$.
   (ii) Explain why there are no integers $x$ and $y$ such that $1547x+504y=10$.

   [6 marks]

   Total for question 4: 12 marks

End of questions