Common divisors, the greatest common divisor (GCD), and the least common multiple (LCM) are essential concepts in mathematics that simplify problems involving fractions, ratios, and number theory. Grasping these concepts is vital for determining relationships between numbers and effectively solving real-world challenges.

Common Divisor

A common divisor is a number that divides two or more numbers exactly, without leaving a remainder. For example, let’s consider the divisors of 28 and 42:

Divisors of 28: 1, 2, 4, 7, 14, 28
Divisors of 42: 1, 2, 3, 6, 7, 14, 21, 42
Common Divisors: 1, 2, 7, 14
GCD: The greatest common divisor is 14.

GCD (Greatest Common Divisor)

The GCD of two or more numbers is the largest number that divides both of them exactly. Mathematically:

GCD(a, b) = The largest common divisor of a and b

Example : Finding the GCD of 48 and 72

Let’s calculate the GCD of 48 and 72 using prime factorization:

Prime Factorization:
48 = 2⁴ × 3
72 = 2³ × 3²
Common Prime Factors: 2 and 3
The smallest power of 2 is 2³, and the smallest power of 3 is 3
GCD Calculation:2³ × 3 = 24
Result: The GCD of 48 and 72 is 24.

LCM (Least Common Multiple)

The LCM of two or more numbers is the smallest number that is a multiple of each of them. Mathematically:

LCM(a, b) = The product of the highest powers of all prime factors of a and b

Example : Finding the LCM of 8 and 12

Let’s calculate the LCM of 8 and 12 using prime factorization:

Prime Factorization:
8 = 2³
12 = 2² × 3
Highest Powers of Prime Factors:
The highest power of 2 is 2³, and the highest power of 3 is 3
LCM Calculation:2³ × 3 = 24
Result: The LCM of 8 and 12 is 24.

Relationship Between GCD and LCM

There is a relationship between the GCD and LCM of two numbers:

GCD(a, b) × LCM(a, b) = a × b

Example : Using the relationship

Let’s consider GCD = 6 and LCM = 60:

GCD × LCM:6 × 60 = 360
If a = 12 and b = 18, then:
a × b:12 × 18 = 360
So, this proves the relationship between GCD and LCM.

Common Divisor and Common Multiple Problems

Example : Finding the GCD and LCM of 50 and 75

Let’s calculate the GCD and LCM of 50 and 75 using prime factorization:

Prime Factorization:
50 = 2 × 5²
75 = 3 × 5²
GCD: The common prime factor is 5²
GCD Calculation:5² = 25
LCM: The highest powers of all prime factors: 2¹, 3¹, 5²
LCM Calculation:2 × 3 × 5² = 150
Result: GCD = 25, LCM = 150

GCD, and LCM Problems

Least Common Multiple and Greatest Common Divisor are commonly used in number theory problems. These types of problems are important for understanding the relationship between numbers and performing the correct operations.

• LCM is used to combine small pieces into larger pieces.
• GCD is used to break down larger pieces into smaller pieces.

Example : Finding the Common Divisors of Two Numbers

Problem: Find all the common divisors of 48 and 60.

Solution:

• Prime factorization:
  • 48 = 2⁴ × 3¹
  • 60 = 2² × 3¹ × 5¹
• Common prime factors:
• Common divisors:

Result: The common divisors are: {1, 2, 3, 4, 6, 12}