Year 7 · Unit 1 Number systems and place value · Lesson 5 of 7

When do two numbers meet?

Two numbers share factors, and their multiples meet again and again. Sort the primes of two numbers into a Venn diagram and read off both answers.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. Write 12 as a product of primes in index form.
    Answer2² × 3.
  2. List the factors of 18.
    Answer1, 2, 3, 6, 9, 18.
  3. List the first five multiples of 4.
    Answer4, 8, 12, 16, 20.
Words used in this lesson
common factor
a factor that two numbers share. For example: 4 is a common factor of 8 and 12.
common multiple
a number that is a multiple of both. For example: 12 and 24 are common multiples of 4 and 6.
greatest common factor (GCF)
the biggest common factor; also called the highest common factor (HCF). For example: the GCF of 8 and 12 is 4.
lowest common multiple (LCM)
the smallest common multiple. For example: the LCM of 4 and 6 is 12.
Venn diagram
overlapping circles that sort what is shared. For example: shared primes sit in the overlap.

Two numbers, shared primes

Last lesson you broke numbers into primes. Now put the primes of two numbers side by side. The primes they share give the greatest common factor (GCF): the biggest number that divides both. Using every prime, each one the greatest number of times it appears in either number, gives the lowest common multiple (LCM): the smallest number that both divide into.

Predict first: 12 and 18 share a 2 and a 3. What is the GCF? And what is the LCM? Then move the sliders, and try multiplying the GCF by the LCM each time.

12
18
6greatest common factor
36lowest common multiple
216 and 216a × b, and GCF × LCM

12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. They share 2 and 3, so the GCF is 6. Every ring in the diagram counted once: 2 × 2 × 3 × 3 = 36 is the LCM.

Show me the steps
  1. a = 12 and b = 18. Shared primes 2 and 3: GCF 6. All primes: 2, 2, 3, 3, so LCM 36.
  2. a = 8 and b = 12. 8 = 2 × 2 × 2 and 12 = 2 × 2 × 3. Shared 2 and 2: GCF 4. LCM = 2 × 2 × 2 × 3 = 24.
  3. a = 7 and b = 10. 7 and 2 × 5 share nothing: GCF 1, and LCM 7 × 10 = 70.
  4. a = 20 and b = 40. 20 is a factor of 40, so the GCF is 20 and the LCM is 40.
  5. In every case GCF × LCM = a × b: 6 × 36 = 216 = 12 × 18.

GCF: multiply the primes in the overlap. It is the largest number that is a factor of both. Some books and exams call it the highest common factor, HCF: it is the same thing.

LCM: multiply every prime in the diagram, once for each ring. It is the smallest number that is a multiple of both.

Always: GCF × LCM = a × b. Use that to check, or to find a missing number. Why: a × b uses every shared prime twice, once from each number. The GCF uses each shared prime once and the LCM uses it once, so together they also use it twice. Every other prime appears once on both sides.

Common multiples as events that line up

Two buses leave a station together, one every 4 minutes and one every 6. When do they next leave together? The common multiples are the times both rows have a dot, and the first is the LCM.

4
6
12first time together
6times together up to 72

Every 4 and every 6: they are together at 12, 24, 36, 48, 60 and 72. The first time is 12, the lowest common multiple.

Show me the steps
  1. Every 4 and every 6: together at 12, 24, 36, 48, 60, 72. The first is 12.
  2. Every 3 and every 5: no shared factor, so they first meet at 3 × 5 = 15, then 30, 45, 60.
  3. Every 6 and every 9: together at 18, 36, 54, 72. The first is 18, not 54.
  4. Every 8 and every 9: they first meet at 72, the end of the timeline.
  5. Every 6 and every 6: the same dots, together every 6, so the first is 6.

A common mistake: “the LCM of 6 and 9 is 6 × 9 = 54.” Multiplying always gives a common multiple, but it is only the lowest one when the numbers share no prime. 6 and 9 both have a 3, so 18 comes first.

Worked example: GCF and LCM from primes

Find the GCF and LCM of 36 and 60.

  1. 36 = 2² × 3² and 60 = 2² × 3 × 5.
  2. Shared primes: 2² and 3. GCF = 2² × 3 = 12.
  3. All primes at their greatest count: 2² × 3² × 5. LCM = 4 × 9 × 5 = 180.
  4. Check: 12 × 180 = 2160 = 36 × 60.

Your go. The first step is given. Find the GCF and LCM of 40 and 56.

  1. 40 = 2³ × 5 and 56 = 2³ × 7.
  2. Find the GCF, then the LCM. Check that GCF × LCM = 40 × 56 before you open the answer.
Check the answer

Shared: 2³, so the GCF is 8. All primes: 2³ × 5 × 7, so the LCM is 280. Check: 8 × 280 = 2240 = 40 × 56.

Check before you go on

1. What is the greatest common factor of 12 and 18?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. What is the greatest common factor of 8 and 12?

3. What is the lowest common multiple of 4 and 6?

Level 2 · Building

4. What is the greatest common factor of 24 and 36?

5. Two buses leave a station together at the start of the day. One leaves every 18 minutes and the other every 30 minutes. After how many minutes do they next leave together?

6. A florist cuts a 48 cm ribbon and a 72 cm ribbon into equal pieces, as long as possible, with none left over. Which is needed, and what is it?

Level 3 · Stretch

7. What is the greatest common factor of 84 and 126?

8. Two numbers have GCF 6 and LCM 72. One of them is 18. What is the other?

Something to think about

Can the greatest common factor of two different numbers be greater than the smaller number? Can the lowest common multiple be smaller than the larger number? When are they equal to it?

One way to think about it

No to both. A factor of a number cannot be more than that number, so the GCF is at most the smaller number, and a multiple of the larger number is at least the larger number, so the LCM is at least the larger. They are equal to those numbers exactly when the smaller number divides the larger, as with 20 and 40.

Exit ticket

9. Karim says the lowest common multiple of 6 and 9 is 54, because 6 × 9 = 54. Which reply is right?