LCM (Least Common Multiple) and GCD (Greatest Common Divisor) are important concepts in number theory used to study relationships between whole numbers, especially factors and multiples.
The Least Common Multiple (LCM) is the smallest positive number that is a multiple of two or more numbers.
The Greatest Common Divisor (GCD), also called the Highest Common Factor (HCF) or Greatest Common Factor (GCF), is the largest positive number that divides two or more numbers exactly without leaving a remainder.
LCM and GCD are widely used in arithmetic, algebra, fractions, measurement, engineering, computer science, scheduling, and many real-life applications involving divisibility and multiples.
A Factor is a whole number that divides another number exactly without leaving a remainder.
(i) The factors of 12 are 1, 2, 3, 4, 6, 12.
(ii) The factors of 18 are 1, 2, 3, 6, 9, 18.
(iii) Since 24 ÷ 6 = 4, 6 is a factor of 24.
A Multiple is the product obtained when a number is multiplied by any whole number.
(i) Multiples of 4: 4, 8, 12, 16, 20, 24, ...
(ii) Multiples of 6: 6, 12, 18, 24, 30, 36, ...
(iii) Since 5 × 7 = 35, 35 is a multiple of 5.
A Prime Number is a whole number greater than 1 that has exactly two factors: 1 and itself.
(i) 2
(ii) 13
(iii) 29
A Composite Number is a whole number greater than 1 that has more than two factors.
(i) 4
(ii) 12
(iii) 30
Common Factors are factors shared by two or more numbers.
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors: 1, 2, 3, 6
Common Multiples are multiples shared by two or more numbers.
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Multiples of 6: 6, 12, 18, 24, 30, ...
Common multiples: 12, 24, 36, ...
The Least Common Multiple (LCM) is the smallest positive number that is exactly divisible by each of the given numbers.
In other words, it is the first common multiple shared by all the given numbers.
The LCM can be found using several methods. The most common methods are:
Find the LCM of 6 and 8.
Step 1: List the multiples of each number.
Multiples of 6:
6, 12, 18, 24, 30, 36, ...
Multiples of 8:
8, 16, 24, 32, 40, ...
Step 2: Identify the first common multiple.
The first common multiple is 24.
Final Answer: LCM = 24
Find the LCM of 18 and 24.
Step 1: Write each number as a product of prime factors.
18 = 2 × 3 × 3 = 2 × 3²
24 = 2 × 2 × 2 × 3 = 2³ × 3
Step 2: Choose each prime factor with its highest power.
Highest power of 2 = 2³
Highest power of 3 = 3²
Step 3: Multiply the selected prime factors.
LCM = 2³ × 3²
= 8 × 9
= 72
Final Answer: LCM = 72
Find the LCM of 12, 18 and 30.
Step 1: Arrange the numbers for short division.
12 18 30
Step 2: Divide by the smallest prime number that divides at least one number.
| 2 | 12 | 18 | 30 |
| 2 | 6 | 9 | 15 |
| 3 | 3 | 9 | 15 |
| 3 | 1 | 3 | 5 |
| 5 | 1 | 1 | 5 |
| 1 | 1 | 1 |
Step 3: Multiply all the divisors.
LCM = 2 × 2 × 3 × 3 × 5
= 180
Final Answer: LCM = 180
Find the LCM of 15 and 20.
Step 1: Prime factorize each number.
15 = 3 × 5
20 = 2² × 5
Step 2: Select each prime factor with the highest power.
LCM = 2² × 3 × 5
Step 3: Multiply the factors.
= 4 × 3 × 5
= 60
Final Answer: LCM = 60
Three buses leave a bus station every 12 minutes, 18 minutes, and 30 minutes respectively. If they all leave the station together at 8:00 a.m., after how many minutes will they leave together again?
Step 1: Identify the numbers.
12, 18 and 30
Step 2: Find their LCM.
Using prime factorization:
12 = 2² × 3
18 = 2 × 3²
30 = 2 × 3 × 5
LCM = 2² × 3² × 5
= 4 × 9 × 5
= 180
Step 3: Interpret the answer.
The buses will leave together again after 180 minutes.
180 minutes = 3 hours.
Therefore, they will leave together again at 11:00 a.m.
Final Answer: 180 minutes (3 hours)
The Greatest Common Divisor (GCD) is the largest positive number that divides two or more numbers exactly without leaving a remainder.
The GCD is also known as the Greatest Common Factor (GCF) or the Highest Common Factor (HCF).
The terms GCD, GCF, and HCF all refer to the same mathematical concept.
A Factor of a number is a whole number that divides the number exactly without leaving a remainder.
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
The common factors are:
1, 2, 3, 6
The greatest common factor is 6.
Therefore, the GCD of 18 and 24 is 6.
The GCD can be found using several methods. The most common methods are:
Find the GCD of 24 and 36.
Step 1: List all the factors of each number.
Factors of 24:
1, 2, 3, 4, 6, 8, 12, 24
Factors of 36:
1, 2, 3, 4, 6, 9, 12, 18, 36
Step 2: Identify the common factors.
1, 2, 3, 4, 6, 12
Step 3: Select the greatest common factor.
The greatest common factor is 12.
Final Answer: GCD = 12
Find the GCD of 48 and 72.
Step 1: Write each number as a product of prime factors.
48 = 2⁴ × 3
72 = 2³ × 3²
Step 2: Choose the common prime factors with the lowest powers.
Common factors:
2³ and 3¹
Step 3: Multiply the selected prime factors.
GCD = 2³ × 3
= 8 × 3
= 24
Final Answer: GCD = 24
Find the GCD of 36, 48, and 60.
Step 1: Arrange the numbers for short division.
36 48 60
Step 2: Divide all the numbers by common prime factors.
| 2 | 36 | 48 | 60 |
| 2 | 18 | 24 | 30 |
| 3 | 9 | 12 | 15 |
| 3 | 4 | 5 |
Step 3: Stop when no common prime factor divides all the numbers.
Step 4: Multiply the common divisors.
GCD = 2 × 2 × 3
= 12
Final Answer: GCD = 12
Find the GCD of 45 and 75.
Step 1: Prime factorize each number.
45 = 3² × 5
75 = 3 × 5²
Step 2: Choose the common prime factors with the lowest powers.
GCD = 3 × 5
Step 3: Multiply the selected factors.
GCD = 15
Final Answer: GCD = 15
A teacher has 48 pencils and 60 rulers. She wants to divide them into the greatest possible number of identical classroom packs so that each pack contains the same number of pencils and the same number of rulers, with none left over. How many packs can she make?
Step 1: Identify the numbers.
48 pencils and 60 rulers
Step 2: Find the GCD of 48 and 60.
48 = 2⁴ × 3
60 = 2² × 3 × 5
GCD = 2² × 3
= 4 × 3
= 12
Step 3: Interpret the answer.
The greatest number of identical classroom packs is 12.
Each pack will contain:
48 ÷ 12 = 4 pencils
60 ÷ 12 = 5 rulers
Final Answer: 12 identical packs, each containing 4 pencils and 5 rulers.
Attempt the following Questions:
(i) Find the Least Common Multiple (LCM) and Greatest Common Divisor (GCD) of the numbers 24, 36, and 60.
(ii) Three traffic lights blink at intervals of 12 seconds, 18 seconds, and 30 seconds respectively. If they all blink together at exactly 8:00 AM, at what time will they next blink together again?
(iii) A farmer has 84 kilograms of apples and 120 kilograms of oranges. She wants to pack them into identical crates such that each crate contains only one type of fruit and every crate has the same maximum possible weight. Find the maximum possible weight of each crate, and determine the total number of crates needed.
(iv) Two ribbons have lengths of 140 cm and 168 cm. They are to be cut into smaller pieces of equal length without any waste. What is the greatest possible length of each piece?
(v) Find the smallest perfect square number that is exactly divisible by 6, 9, and 15.
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