Buses
Two buses leave a station together at 9:00. One leaves every 12 minutes and the other every 18 minutes. Work out when they next leave together.
36 minutes later, at 9:36
Before this page: find the highest common factor and the lowest common multiple of two numbers from their prime factorisations, or from a Venn diagram.
The same two numbers can ask for a multiple or for a factor
Ribbons
Two ribbons are 12 cm and 18 cm long. Both are cut into pieces of the same length, as long as possible, with nothing left over. Work out the length of a piece.
6 cm
Both questions give 12 and 18. In the bus question the numbers are steps, every 12 minutes and every 18. The buses leave at the multiples of 12 and of 18, so they are together again at a common multiple, and the first is the lowest common multiple, 36.
In the ribbon question the numbers are whole amounts. A piece that leaves nothing over is a factor of 12 and of 18, so the longest piece is the highest common factor, 6.
Steps that line up need the LCM; amounts split into equal pieces need the HCF
- The numbers are steps: every 12 minutes, packs of 8, the teeth on a gear. The answer is a multiple of both, so it is at least as large as the larger number.
- The numbers are whole amounts: 18 cm of ribbon, 30 pens. The answer is a factor of both, so it is no larger than the smaller number.
The size of the answer tests the choice: 6 minutes would come before either bus had left again, and a 36 cm piece would be longer than either ribbon.
Repeating events are together again at the lowest common multiple
Buses every 12 minutes and every 18 leave together at every common multiple of 12 and 18: the first is the LCM, and the later ones are its multiples.
Buses every 12 minutes and every 18 minutes
Both buses leave at 9:00
One bus leaves every 12 minutes and the other every 18: each row of the timetable is a count in one step.
Gears and packs also meet at a common multiple
Gear teeth meet again after both tooth counts divide the number that has passed. Packs match when both pack sizes divide the number of items bought. The first possible total is the LCM.
Example: two gears
A gear with 20 teeth drives a gear with 12 teeth. One tooth on each gear is marked, and the two marks are touching. Work out how many turns the 20-tooth gear makes before the marks touch again.
- The numbers are steps: a turn of the large gear passes 20 teeth, and a turn of the small gear passes 12. The marks meet when the same number of teeth has passed on both gears, a multiple of 20 and of 12; the first such number is the LCM.
- LCM(20, 12) = 60 20 = 22 × 5 and 12 = 22 × 3; every prime to its higher index gives 22 × 3 × 5 = 60.
- 60 ÷ 20 = 3 turns 60 teeth pass the meeting point, and the large gear has 20, so it turns 3 times; the small gear turns 60 ÷ 12 = 5 times.
Example: packs that match
Hot dogs come in packs of 10 and rolls in packs of 8. Work out the smallest number of each pack that gives the same number of hot dogs as rolls.
- The numbers are steps: 10 hot dogs a pack and 8 rolls a pack. Only whole packs are bought, so the number of hot dogs is a multiple of 10 and the number of rolls a multiple of 8; the smallest number that is both is the LCM.
- LCM(10, 8) = 40 10 = 2 × 5 and 8 = 23; 23 × 5 = 40.
- 40 ÷ 10 = 4 packs of hot dogs and 40 ÷ 8 = 5 packs of rolls 40 is the number of hot dogs and of rolls; the packs are how many 10s and how many 8s make 40.
Equal pieces with nothing left over come from the highest common factor
A 60 cm by 36 cm card is cut into equal squares with nothing over. Squares fill the 60 cm across only if their side divides 60, and fill the 36 cm down only if it divides 36: the side is a common factor of the two lengths. The largest square has the largest common factor as its side.
Cutting a 60 cm by 36 cm card into equal squares
A 60 cm by 36 cm card
60 cm across and 36 cm down, to be cut into squares of one size with nothing left over.
Example: identical packs
30 pens and 42 pencils are put into packs, every pack the same, with nothing left over. Work out the greatest number of packs and what each pack holds.
- The numbers are whole amounts: 30 pens and 42 pencils are shared out. Every pack holds the same number of pens, so the number of packs divides 30, and it divides 42 the same way; the greatest number dividing both is the HCF.
- HCF(30, 42) = 6 30 = 2 × 3 × 5 and 42 = 2 × 3 × 7; the shared primes give 2 × 3 = 6.
- 6 packs, each holding 30 ÷ 6 = 5 pens and 42 ÷ 6 = 7 pencils 5 and 7 share no prime, so the 6 packs cannot be split into more.
The LCM or HCF is one step from the answer the question asks for
The LCM or HCF is a fact about the two numbers; the question is about buses, gears, packs or a card, and one more step turns the value into the answer asked for.
| The question | The numbers | Needs | Value | The answer asked for |
|---|---|---|---|---|
| Buses every 12 and 18 minutes | steps | LCM | 36 | 9:36 |
| Gears with 20 and 12 teeth | steps | LCM | 60 | 60 ÷ 20 = 3 turns |
| Packs of 10 and packs of 8 | steps | LCM | 40 | 4 packs and 5 packs |
| A 60 cm by 36 cm card | amounts | HCF | 12 | 12 cm squares, 5 × 3 = 15 of them |
| 30 pens and 42 pencils | amounts | HCF | 6 | 6 packs of 5 pens and 7 pencils |
Two numbers have both answers, and the question chooses between them
When one number is a multiple of the other, as with 12 and 36, both answers are among the numbers given. Buses every 12 and every 36 minutes are together every 36 minutes, LCM(12, 36) = 36; ribbons of 12 cm and 36 cm cut into 12 cm pieces, HCF(12, 36) = 12.
Steps of 15 and steps of 20
15 and 20 split into equal pieces
Common mistakes
- Multiplying the two numbers in a bus question 12 × 18 = 216 minutes is a time both buses leave, but 36 comes first. 12 and 18 share the factor 6, so their product is a common multiple and not the lowest; the product is the LCM only when the two numbers share no prime.
- Answering the card question with the LCM LCM(60, 36) = 180, and a 180 cm square does not fit on a 60 cm by 36 cm card. A piece cut from an amount is never larger than the amount: the side is a factor of both lengths, 12.
- Stopping at the LCM when the question asks for packs 40 is the number of hot dogs and the number of rolls, not a number of packs. The packs are 40 ÷ 10 = 4 and 40 ÷ 8 = 5.
- Giving the HCF as the number of squares Answering 12 squares gives the side of one square as the count. 12 cm is the side, and the count is (60 ÷ 12) × (36 ÷ 12) = 5 × 3 = 15.
Summary
- Numbers that are steps, like every 12 minutes or packs of 8, line up at a common multiple; the first time is the LCM.
- Numbers that are whole amounts split into equal pieces with nothing left over need a common factor, and the largest piece is the HCF.
- An LCM is at least as large as the larger number and an HCF no larger than the smaller: an answer of the wrong size is the wrong choice.
- Any two numbers have both an LCM and an HCF, and the question chooses between them; when one number is a multiple of the other, both are among the numbers given.
- The LCM or HCF is then turned into what the question asked for: a clock time, a number of turns, a number of packs or pieces, or what each pack holds.