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 Introduction – What is an Algorithm?

Basic Definition:

An algorithm is a step-by-step method used to solve a problem in a logical and organized way.

Set of rules – A group of rules that help complete a calculation, either by hand or using a machine.

Well-defined computational procedure – A method that takes input, processes it, and gives an output.

Finite sequence of steps – A process with a fixed number of steps to reach a result.

Where does the word "Algorithm" come from?

The word algorithm comes from the name of a 9th-century mathematician,
Abdullah Jafar Muhammad ibn Musa Al-Khwarizmi.

 What an Algorithm is NOT:

• An algorithm is not a programming language.
• It is not actual code, but a step before coding.
• It is usually written in simple language, so anyone can understand it.

Key Properties of a Good Algorithm:

Property		Explain
Input		It should have a fixed number of inputs.
Output		It must give at least one output.
Definiteness		Every step must be clear and precise.
Effectiveness		Steps should be simple and effective.
Finiteness		It must finish in a limited number of steps.

Useful Qualities of a Great Algorithm:

General – It should work for more than one specific problem.

Efficient – Uses less time and less memory.

Understandable – Easy for anyone to read and follow.

Unambiguous – Each step should be clear and not confusing.

What is GCD (Greatest Common Divisor)?

The GCD of two numbers is the largest number that completely divides both numbers.

Example:
GCD of 8 and 12 is 4,
because 4 divides both 8 and 12, and no bigger number does.

What is Euclid’s Algorithm?

Euclid’s Algorithm is a very old and efficient method to find the GCD of two numbers.
It was invented by the Greek mathematician Euclid.

Steps of Euclid’s Algorithm (in Simple Words):

Input: Two positive integers a and b where a > b

1. If b = 0, then a is the GCD → Stop

2. Otherwise:

   • Find the remainder r = a mod b

   • Replace:
     a = b
b = r

   • Go back to Step 1

Repeat this process until the remainder becomes 0.

Let’s Understand with an Example:

Find GCD of 1071 and 462

Iteration	a	b	r = a mod b	Next Step (a ← b, b ← r)
1	1071	462	147	a = 462, b = 147
2	462	147	21	a = 147, b = 21
3	147	21	0	Done!

When b = 0, we stop.
Final Answer: GCD = 21

Pseudo-code of Euclid’s Algorithm:

Algorithm GCD-Euclid(a, b)

begin

   while b ≠ 0 do

      r ← a mod b

      a ← b

      b ← r

   return a

end

Key Points:

• Inputs: Two positive integers a and b

• Output: GCD of a and b

• Terminates: Always finishes in finite steps

• Efficient: Very fast, even for large numbers

Real-Life Uses of GCD:

• To simplify fractions
  (e.g., 18/24 becomes 3/4 after dividing both by GCD = 6)

• In Cryptography, especially RSA algorithm

• In Number Theory and Math competitions