GCF and LCM — Print
Prime factorisation answers both questions at once: shared factors give the GCF, and everything needed gives the LCM.
What the practice questions cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Factor — A number that divides evenly into another.
- Multiple — The result of multiplying by a whole number: 12 is a multiple of 4.
- GCF — Greatest common factor — the largest number dividing both.
- LCM — Least common multiple — the smallest number both divide into.
- Prime factorisation — A number written as a product of primes: 60 = 22 · 3 · 5.
Two gears mesh together: one with 16 teeth, one with 12. Mark one tooth on each with paint, and line the marks up.
Now turn them. How far must they go before both marks are back at the top together?
The big gear returns every 16 teeth. The small one every 12. You need a number that both reach.
24? The small one is home, the big one is not. 48 works: 16 × 3 and 12 × 4.
That is the lowest common multiple — the first moment two repeating cycles agree.
The GCF is the opposite question: not when do they next agree, but what is the largest piece they share. Here 4 — you could split both gears into groups of four teeth, and no larger group works for both.
Factors go down, multiples go up
Factors of 12: 1, 2, 3, 4, 6, 12 — a short, finite list, all no bigger than 12.
Multiples of 12: 12, 24, 36, 48, … — an endless list, all no smaller than 12.
Mixing these two up is the root of most GCF/LCM errors, so fix the direction first: GCF is small, LCM is large.
Prime factorisation does both jobs
Break each number into primes with a factor tree. 24 = 23 · 3 and 36 = 22 · 32.
GCF: take each shared prime to the lowest power. Here 22 · 3 = 12.
LCM: take every prime that appears, to the highest power. Here 23 · 32 = 72.
Lowest for GCF, highest for LCM
Both use the same factorisations. The GCF keeps only what the numbers share; the LCM keeps everything either one needs.
A useful check
GCF × LCM = a × b, always.
For 24 and 36: 12 × 72 = 864, and 24 × 36 = 864 ✓
This catches mistakes instantly, and it lets you find one if you already know the other.
Which one does the word problem want?
GCF when you are splitting things into equal groups, or cutting into equal pieces as large as possible. The answer is smaller than the numbers.
LCM when two cycles start together and you want the next time they line up, or when you are buying packs of different sizes to match. The answer is larger.
Ask: bigger or smaller?
Before computing anything, decide whether the answer should be bigger or smaller than the numbers in the problem. That single question picks GCF or LCM correctly almost every time.
Relatively prime numbers
If two numbers share no factor but 1, their GCF is 1 and they are called relatively prime — like 8 and 15.
In that case the LCM is simply their product: 8 × 15 = 120.
Worked examples
Both at once
Find the GCF and LCM of 18 and 30.
- Factorise: 18 = 2 · 32 and 30 = 2 · 3 · 5 Prime factorisation makes both answers readable.
- GCF: shared primes at lowest powers — 2 · 3 = 6 Both have one 2 and at least one 3; neither shares a 5.
- LCM: every prime at highest power — 2 · 32 · 5 = 90 Take 32 from 18 and the 5 from 30.
- Check: 6 × 90 = 540 = 18 × 30 ✓ The product rule confirms both.
GCF = 6, LCM = 90.
Which one?
Two lighthouses flash every 12 and 18 seconds. They flash together now. When do they next flash together?
- Ask: bigger or smaller than 12 and 18? The next shared flash is later, so bigger — that means LCM.
- 12 = 22 · 3, 18 = 2 · 32 Factorise both.
- LCM = 22 · 32 = 36 Highest power of each prime.
In 36 seconds.
The other kind
A florist has 48 roses and 36 tulips, and wants identical bunches using every flower. What is the largest number of bunches?
- Ask: bigger or smaller? The number of bunches must divide both counts, so smaller — GCF.
- 48 = 24 · 3 and 36 = 22 · 32 Factorise.
- GCF = 22 · 3 = 12 Lowest power of each shared prime.
- Each bunch: 4 roses and 3 tulips. 48 ÷ 12 and 36 ÷ 12 — worth stating, as it is often the follow-up question.
12 bunches.
Words to know
- Factor
- A number that divides evenly into another.
- Multiple
- The result of multiplying by a whole number: 12 is a multiple of 4.
- GCF
- Greatest common factor — the largest number dividing both.
- LCM
- Least common multiple — the smallest number both divide into.
- Prime factorisation
- A number written as a product of primes: 60 = 22 · 3 · 5.
Mistakes to watch for
Swapping them. The GCF can never be larger than the smaller number, and the LCM can never be smaller than the larger one. That one sanity check catches the swap immediately.
Using the highest powers for the GCF. It is the other way round: GCF takes the lowest power of each shared prime, LCM the highest. The GCF must divide both, so it cannot take more than either has.
Forgetting a prime that appears in only one number when building the LCM. The LCM must be a multiple of both, so every prime in either number has to be in it.
Reaching for the formula when the question is about timing or sharing. "When do they coincide?" is LCM; "what is the biggest equal group?" is GCF. Decide which story you are in first.
The bicycle world calls a gear pair with a common factor a non-hunting gear — the same teeth meet each other every cycle, so wear concentrates on those teeth.
Engineers deliberately choose tooth counts with no common factor, often a prime, so every tooth eventually meets every other and wear spreads evenly.
Some cicadas emerge every 13 or 17 years. Both are prime, which makes it hard for any predator with a shorter cycle to sync up with them.
Problem 1
Find the GCF of 12 and 18.
Worked answer
6
Factors of 12: 1,2,3,4,6,12
Factors of 18: 1,2,3,6,9,18
Largest shared: 6
Problem 2
Find the LCM of 4 and 6.
Worked answer
12
Multiples of 6: 6, 12, …
12 is the first also divisible by 4.
Problem 3
What is the prime factorisation of 36?
- 22 · 32
- 23 · 3
- 4 · 9
- 62
Worked answer
22 · 32
36 = 4 × 9 = 22 · 32
4, 9 and 6 are not prime, so those answers are incomplete.
Problem 4
Find the GCF of 8 and 15.
Worked answer
1
8 = 23 and 15 = 3 · 5
No shared primes, so the GCF is 1 — they are relatively prime.
Problem 5
Find the LCM of 5 and 7.
Worked answer
35
5 and 7 are relatively prime
LCM = 5 × 7 = 35
Problem 6
Find the GCF of 24 and 36.
Worked answer
12
24 = 23 · 3, 36 = 22 · 32
Shared at lowest powers: 22 · 3
= 12
Problem 7
Find the LCM of 12 and 18.
Worked answer
36
12 = 22 · 3, 18 = 2 · 32
Highest powers: 22 · 32 = 36
Problem 8
Find the GCF of 18 and 30.
Worked answer
6
18 = 2 · 32, 30 = 2 · 3 · 5
Shared: 2 · 3 = 6
Problem 9
Find the LCM of 8 and 12.
Worked answer
24
Highest powers: 23 · 3
= 24
Problem 10
Two lighthouses flash every 12 and 18 seconds, together right now. After how many seconds do they next flash together?
Worked answer
36
LCM of 12 and 18
22 · 32 = 36 seconds
Problem 11
A florist has 48 roses and 36 tulips and wants identical bunches using every flower. What is the greatest number of bunches?
Worked answer
12
48 = 24 · 3, 36 = 22 · 32
GCF = 22 · 3 = 12
Each bunch: 4 roses and 3 tulips.
Problem 12
Two numbers have GCF 6 and LCM 90. One of them is 18. Find the other.
Worked answer
30
6 × 90 = 540
540 ÷ 18 = 30
Check: GCF(18, 30) = 6 ✓
Problem 13
Find the GCF of 84 and 126.
Worked answer
42
84 = 22 · 3 · 7
126 = 2 · 32 · 7
Shared at lowest powers: 2 · 3 · 7 = 42
Problem 14
Find the LCM of 15, 20 and 25.
Worked answer
300
15 = 3 · 5, 20 = 22 · 5, 25 = 52
Highest powers: 22 · 3 · 52
= 4 × 3 × 25 = 300
Problem 15
Two numbers have GCF 8 and LCM 48. One is 16. Find the other.
Worked answer
24
8 × 48 = 384
384 ÷ 16 = 24
Problem 16
What is the smallest number divisible by 6, 8 and 15?
Worked answer
120
6 = 2 · 3, 8 = 23, 15 = 3 · 5
Highest powers: 23 · 3 · 5
= 120