FactorsHCF and LCM word problems, solved
The hardest part of these questions is not the calculation. It is deciding whether the question wants the HCF or the LCM. This page gives a simple test and ten solved problems of the kinds set in classes 5 to 10.
HCF or LCM? A quick test
| The question is about | Use | The answer is |
|---|---|---|
| Splitting, cutting, sharing or measuring exactly; the largest, longest, greatest | HCF | Smaller than or equal to the numbers given |
| Things repeating and meeting again; the smallest, least, earliest | LCM | Larger than or equal to the numbers given |
Before calculating, ask: should my answer be smaller than the numbers in the question, or larger? That one thought settles most questions.
HCF problems
1. The longest tape
Two ropes are 84 m and 126 m long. What is the greatest length of tape that can measure both exactly?
84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. The HCF is 2 × 3 × 7 = 42. The tape is 42 m long.
2. Equal packets
A shopkeeper has 48 pens and 72 pencils. He wants identical packets using all of them. What is the greatest number of packets?
HCF(48, 72) = 24. He can make 24 packets, each with 2 pens and 3 pencils.
3. Square tiles
A floor is 540 cm by 360 cm. What is the largest square tile that covers it without cutting, and how many are needed?
HCF(540, 360) = 180. The tile is 180 cm square. The floor takes 3 tiles one way and 2 the other: 6 tiles.
4. Given remainders
Find the largest number that divides 70 and 125 leaving remainders 5 and 8.
Take the remainders away first: 70 − 5 = 65 and 125 − 8 = 117. The number must divide both 65 and 117 exactly. HCF(65, 117) = 13. The answer is 13. Check: 70 = 13 × 5 + 5 and 125 = 13 × 9 + 8.
5. Rows in a parade
Two groups of 32 and 616 members march behind each other in the same number of columns. What is the greatest number of columns?
HCF(616, 32): 616 ÷ 32 leaves 8, and 32 ÷ 8 leaves 0. The answer is 8 columns.
LCM problems
6. Bells
Three bells ring every 9, 12 and 15 minutes. They ring together at 8:00. When do they next ring together?
LCM(9, 12, 15) = 2² × 3² × 5 = 180 minutes, which is 3 hours. They ring together again at 11:00.
7. Traffic lights
Lights change every 48, 72 and 108 seconds. They change together at 7:00:00. When next?
LCM(48, 72, 108) = 2⁴ × 3³ = 432 seconds, which is 7 minutes 12 seconds. They change together at 7:07:12.
8. Smallest number with a remainder
Find the smallest number which leaves remainder 7 when divided by 12, 16 and 24.
LCM(12, 16, 24) = 48. Add the remainder: 48 + 7 = 55. Check: 55 = 12 × 4 + 7 = 16 × 3 + 7 = 24 × 2 + 7.
9. Round the track
Two runners take 18 minutes and 12 minutes for one round. They start together. After how long do they meet at the start again?
LCM(18, 12) = 36 minutes. By then one has run 2 rounds and the other 3.
10. Using both
The HCF of two numbers is 9 and their LCM is 360. One number is 45. Find the other.
For two numbers, HCF × LCM = product. So the other number is 9 × 360 ÷ 45 = 72. Check: HCF(45, 72) = 9 and LCM(45, 72) = 360.
Checking your answer
- An HCF must divide every number in the question. Divide and see.
- An LCM must be divisible by every number in the question.
- If your "HCF" is bigger than the numbers, or your "LCM" is smaller, you chose the wrong one.
Work any of these through with the HCF and LCM calculator, which shows each step. For printed practice with answers, use the HCF and LCM worksheet maker on our Worksheets site.