GREATEST COMMON DIVISOR (GCD) AND LEAST COMMON MULTIPLE (LCM)
Definitions
Greatest Common Divisor (GCD):
The greatest common divisor refers to the greatest positive integer that is a common divisor of a given set of positive integers.
It is also known as the Highest Common Factor (HCF) or Greatest Common Factor (GCF).
Least Common Multiple (LCM):
The least common multiple is the smallest positive number that is a multiple of two or more numbers.
A multiple of a number is obtained by multiplying it by 1, 2, 3, 4, 5, and so on.
Rules and Shortcuts
| Situation | Use |
|---|---|
| Splitting items into equal largest parts | GCD |
| Finding smallest number for repeated events | LCM |
| Answer smaller than given numbers | GCD |
| Answer larger than given numbers | LCM |
| Key words: Greatest, Largest | GCD |
| Key words: Least, Smallest, Shortest | LCM |
Example 1.9
Find the GCD of 12, 36 and 56.
12 = 1, 2, 3, 4, 6, 12
36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
56 = 1, 2, 4, 7, 8, 14, 28, 56
Common factors: 1, 2, 4
GCD(12,36,56) = 4
Example 1.10
Find the LCM of 6, 12 and 16.
Multiples of 6: 6,12,18,24,30,36,42,48…
Multiples of 12: 12,24,36,48…
Multiples of 16: 16,32,48…
Smallest common multiple = 48
LCM(6,12,16) = 48
Application of LCM and GCD
Example
A room measures 540 cm by 420 cm. Determine the length of the largest square tiles that can be used to cover the floor without requiring cutting.
Largest tile → GCD(540,420)
GCD = 60 cm
Example
Find the shortest length of a thread that can be cut into pieces of 14 cm or 18 cm without leaving remainders.
Shortest length → LCM(14,18)
LCM = 126 cm
Exercise 1.7
1. Find the GCD of the following sets of numbers
(a) 72, 108 and 156
Prime factors → GCD = 12
(b) 90, 145 and 150
Common factor = 5
(c) 54, 90 and 108
GCD = 18
(d) 30, 42 and 96
GCD = 6
(e) 66, 88 and 176
GCD = 22
(f) 88, 96 and 152
GCD = 8
2. Find the LCM of each of the following
(a) 15, 18 and 30
LCM = 90
(b) 9, 12 and 30
LCM = 180
(c) 18, 24 and 30
LCM = 360
(d) 16, 18 and 72
LCM = 144
(e) 12, 24 and 36
LCM = 72
(f) 15, 33 and 45
LCM = 495
3.
Six bells commence ringing together at intervals of 3, 6, 9, 12, 15 and 18 seconds respectively.
In 30 minutes, how many times do they ring together?
LCM = 18 seconds → 30 minutes = 1800 seconds → 1800 ÷ 180 = 10
4.
Find the least number which leaves a remainder of 3 when divided by 8, 16, 24 and 48.
LCM = 48 → 48 + 3 = 51
5.
Three friends J, K and L jog around a circular stadium and complete one round in 12, 18 and 20 seconds respectively.
In how many minutes will they all meet again at the starting point?
LCM = 180 seconds = 3 minutes
6.
Three cloths of 700 cm, 750 cm and 950 cm respectively have to be cut into same length pieces of maximum size.
What will be the number of pieces?
GCD=50, pieces=14+15+19=48