GCF and LCM — Practice
Prime factorisation answers both questions at once: shared factors give the GCF, and everything needed gives the LCM.
What these 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.
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