The greatest common factor is the largest number that divides every one of your numbers exactly. For 24, 36 and 60 it is 12: the primes all three share, at their lowest power, are 2² × 3. On site it gives the largest square tile, panel or equal piece that fits two or more lengths with no cuts or waste.
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Greatest common factor (GCF / HCF)
12
Common factors
1, 2, 3, 4, 6, 12
24 in prime factors
2³ × 3
36 in prime factors
2² × 3²
60 in prime factors
2² × 3 × 5
GCF in prime factors
2² × 3
Least common multiple (LCM)
360
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Formula
Write each number as prime factorsGCF = only the primes in every number, at their lowest powerEuclid: divide, keep the remainder, repeat; the last divisor is the GCFTwo numbers: GCF = a × b ÷ LCM
Worked Example
Find the GCF of 24, 36 and 60:
24 = 2³ × 3, 36 = 2² × 3², 60 = 2² × 3 × 5
Primes in all three: 2 (lowest power 2²) and 3 (lowest power 3¹); 5 is only in 60
Largest square that fits both sides: 3 × 4 = 12 panels
Pieces from two boards
240 cm and 300 cm
60 cm
Longest equal piece with no offcut: 4 + 5 = 9 pieces
Posts along two sides
12 m and 18 m
6 m
Widest equal spacing with a post at every corner; 3 m or 2 m also fit
Simplifying a ratio
300 : 450
150
Divide both by 150: 2 : 3
GCF of Common Pairs
Numbers
GCF
LCM
8 and 12
4
24
12 and 18
6
36
15 and 25
5
75
16 and 24
8
48
18 and 24
6
72
20 and 30
10
60
24 and 36
12
72
36 and 48
12
144
48 and 18
6
144
60 and 90
30
180
120 and 180
60
360
360 and 480
120
1,440
Frequently Asked Questions
How do you find the greatest common factor?
Write each number in prime factors, then multiply only the primes that appear in every number, each at its lowest power. 24 = 2³ × 3, 36 = 2² × 3² and 60 = 2² × 3 × 5, so the GCF is 2² × 3 = 12.
What is the difference between GCF, HCF and GCD?
There is none. Greatest common factor (GCF), highest common factor (HCF, the usual term in the UK and India) and greatest common divisor (GCD) all mean the same number.
How is the GCF used in construction?
To find the largest equal size that fits several lengths with no waste. A 3.6 × 4.8 m room takes whole 1.2 m square panels, because the GCF of 360 and 480 cm is 120 cm: 3 × 4 = 12 panels.
What is Euclid's method?
Divide the larger number by the smaller and keep the remainder, then divide the last divisor by that remainder, until the remainder is 0. For 48 and 18: 48 = 2 × 18 + 12, 18 = 1 × 12 + 6, 12 = 2 × 6 + 0, so the GCF is 6.
What is the GCF of 12 and 18?
6. 12 = 2² × 3 and 18 = 2 × 3², and the primes they share, at their lowest power, are 2 × 3 = 6.
What does a GCF of 1 mean?
The numbers share no factor except 1; they are called coprime, for example 8 and 15. Equal whole pieces then fit both only if each piece is 1 unit long.
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