LCM/HCF

Number

These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.

Find the highest common factor of $18$ and $24$.

2 marks
Find the lowest common multiple of $8$ and $12$.

2 marks
Find the HCF and the LCM of $20$ and $30$.

3 marks
Bus A leaves a station every $15$ minutes and bus B leaves every $20$ minutes. Both leave at $08:00$. When do they next leave together?

3 marks
Two numbers have HCF $6$ and LCM $90$. One of the numbers is $18$. Find the other number.

4 marks
Two ribbons are $84$ cm and $126$ cm long. They are cut into the longest possible equal pieces, with no ribbon left over. How long is each piece, and how many pieces are there altogether?

4 marks
Find the smallest positive integer that is a multiple of $12$, $18$ and $20$.

5 marks
The HCF of two numbers is $8$ and their LCM is $96$. Both numbers are between $20$ and $50$. Find the two numbers.

6 marks

Mark scheme

Answer. $6$
- Factors of $18$ include $1, 2, 3, 6, 9, 18$.
- The greatest factor shared with $24$ is $6$.
Answer. $24$
- Multiples of $8$: $8, 16, 24, \ldots$
- The first one also in the multiples of $12$ is $24$.
Answer. HCF $10$, LCM $60$
- $20 = 2^{2} \times 5$ and $30 = 2 \times 3 \times 5$.
- HCF $= 2 \times 5 = 10$.
- LCM $= 2^{2} \times 3 \times 5 = 60$.
Answer. $09:00$
- LCM of $15$ and $20$ is $60$.
- $60$ minutes after $08:00$ is $09:00$.
Answer. $30$
- Product of the two numbers $= 6 \times 90 = 540$.
- $540 \div 18 = 30$.
- Check: HCF of $18$ and $30$ is $6$, and LCM is $90$.
Answer. $42$ cm, and $5$ pieces
- $84 = 2^{2} \times 3 \times 7$ and $126 = 2 \times 3^{2} \times 7$.
- HCF $= 2 \times 3 \times 7 = 42$ cm.
- Number of pieces $= 84 \div 42 + 126 \div 42 = 5$.
Answer. $180$
- $12 = 2^{2} \times 3$, $18 = 2 \times 3^{2}$, $20 = 2^{2} \times 5$.
- LCM $= 2^{2} \times 3^{2} \times 5$.
- $4 \times 9 \times 5 = 180$.
Answer. $24$ and $32$
- The product of the numbers is $8 \times 96 = 768$.
- Factor pairs of $768$ that are multiples of $8$ include $24$ and $32$.
- HCF$(24, 32) = 8$ and LCM$(24, 32) = 96$.
- Both are between $20$ and $50$. The pair $16$ and $48$ is rejected because $16$ is too small.