HCF and LCM: solved examples

Worked examples of hcf and lcm, from easy to hard, with every step written out. Try each one yourself first, then check the working.

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Easy

Example 1. What is the highest common factor (HCF) of $24$ and $54$ ?
1. Find a big number that divides both. $24 = 6 \times 4, \quad 54 = 6 \times 9$
2. 4 and 9 have no common factor, so the HCF is 6. $\operatorname{HCF}(24, 54) = 6$
Answer: $6$
Example 2. What is the highest common factor (HCF) of $6$ and $15$ ?
1. Find a big number that divides both. $6 = 3 \times 2, \quad 15 = 3 \times 5$
2. 2 and 5 have no common factor, so the HCF is 3. $\operatorname{HCF}(6, 15) = 3$
Answer: $3$
Example 3. What is the highest common factor (HCF) of $81$ and $18$ ?
1. Find a big number that divides both. $81 = 9 \times 9, \quad 18 = 9 \times 2$
2. 9 and 2 have no common factor, so the HCF is 9. $\operatorname{HCF}(81, 18) = 9$
Answer: $9$

Example 4. What is the lowest common multiple (LCM) of $24$ and $54$ ?
1. Find the HCF first. $\operatorname{HCF}(24, 54) = 6$
2. Then LCM = a × b ÷ HCF. $\operatorname{LCM} = \frac{24 \times 54}{6} = 216$
Answer: $216$
Example 5. What is the lowest common multiple (LCM) of $6$ and $15$ ?
1. Find the HCF first. $\operatorname{HCF}(6, 15) = 3$
2. Then LCM = a × b ÷ HCF. $\operatorname{LCM} = \frac{6 \times 15}{3} = 30$
Answer: $30$
Example 6. What is the lowest common multiple (LCM) of $81$ and $18$ ?
1. Find the HCF first. $\operatorname{HCF}(81, 18) = 9$
2. Then LCM = a × b ÷ HCF. $\operatorname{LCM} = \frac{81 \times 18}{9} = 162$
Answer: $162$

Example 7. One bus leaves every $24$ minutes and another every $54$ minutes. They leave together at 9:00. After how many minutes do they next leave together?
1. Find the HCF first. $\operatorname{HCF}(24, 54) = 6$
2. Then LCM = a × b ÷ HCF. $\operatorname{LCM} = \frac{24 \times 54}{6} = 216$
Answer: $216$
Example 8. A floor is $6$ cm by $15$ cm. What is the side of the largest square tile that covers it exactly, with no cutting? (in cm)
1. Find a big number that divides both. $6 = 3 \times 2, \quad 15 = 3 \times 5$
2. 2 and 5 have no common factor, so the HCF is 3. $\operatorname{HCF}(6, 15) = 3$
Answer: $3$
Example 9. One bus leaves every $81$ minutes and another every $18$ minutes. They leave together at 9:00. After how many minutes do they next leave together?
1. Find the HCF first. $\operatorname{HCF}(81, 18) = 9$
2. Then LCM = a × b ÷ HCF. $\operatorname{LCM} = \frac{81 \times 18}{9} = 162$
Answer: $162$

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