GREATEST COMMON DIVIDER: APPLICATION EXERCISES

The greatest common factor (GCD) is the largest integer that exactly divides two or more integers. It is also known as the "greatest common factor" or "greatest common divisor."

formal definition

The GCD of two integers a and b is the largest integer that exactly divides a and b.

MCD Properties

1. _Uniqueness_: The GCF of two integers is unique.
2. _Divisibility_: The GCD of two integers divides both numbers exactly.
3. _Maximum_: The GCD is the largest integer that exactly divides both numbers.
4. _Associativity_: The GCF of three integers a, b and c is equal to the GCD of the GCF of a and b, and c.

Methods to find the GCD

1. _Factorization method_: Factor the integers into their prime factors and find the product of the common factors.
2. _Euclid's Method_: Use Euclid's algorithm to find the GCF of two integers.
3. _Subtraction method_: Subtract the smallest number from the largest number until you obtain a remainder of 0. The last remainder other than 0 is the GCF.

Examples

1. Find the GCD of 12 and 18:
- Factorization: 12 = 2^2 × 3, 18 = 2 × 3^2
- GCD: 2 × 3 = 6
1. Find the GCF of 24 and 30:
- Factorization: 24 = 2^3 × 3, 30 = 2 × 3 × 5
- GCD: 2 × 3 = 6

MCD Applications

1. _Number theory_: The GCF is used in number theory to study the properties of integers.
2. _Algebra_: The GCD is used in algebra to find the greatest common divisor of two polynomials.
3. _CRYPTOGRAPHY_: The MCD is used in cryptography to develop encryption and decryption algorithms.
4. _Computer Science_: MCD is used in computer science to develop data processing algorithms.