Why GCD and LCM Euclidean Algorithm in DSA Typescript?

The Big Idea

Discover how a simple remainder trick can solve big number puzzles in seconds!

The Scenario

Imagine you want to find the greatest common divisor (GCD) or least common multiple (LCM) of two big numbers by listing all their factors or multiples on paper.

This can take forever and is very tiring!

The Problem

Manually listing factors or multiples is slow and easy to mess up, especially with large numbers.

You might miss some factors or get confused, leading to wrong answers.

The Solution

The Euclidean Algorithm uses a simple trick: it replaces the big problem with smaller ones by using remainders.

This way, it quickly finds the GCD without listing all factors.

Once you have the GCD, you can easily find the LCM with a simple formula.

Before vs After

✗ Before

function gcdManual(a: number, b: number): number {
  // List all factors and find the biggest common one
  // Very slow and complex for big numbers
  return 1; // placeholder
}

✓ After

function gcdEuclidean(a: number, b: number): number {
  while (b !== 0) {
    const remainder = a % b;
    a = b;
    b = remainder;
  }
  return a;
}

What It Enables

You can quickly and reliably find GCD and LCM of very large numbers, enabling fast math calculations and problem solving.

Real Life Example

When you want to schedule two repeating events so they happen together as often as possible, you use LCM.

The Euclidean Algorithm helps find that LCM fast, even if the events repeat every big number of days.

Key Takeaways

Manual factor listing is slow and error-prone.

Euclidean Algorithm uses remainders to find GCD quickly.

LCM can be found easily once GCD is known.

Practice

1. What does the Euclidean algorithm primarily calculate between two numbers?

easy

A. The difference between the two numbers

B. The Least Common Multiple (LCM)

C. The Greatest Common Divisor (GCD)

D. The sum of the two numbers

Solution

1. Step 1: Understand the Euclidean algorithm purpose

   The Euclidean algorithm is a method to find the greatest common divisor (GCD) of two integers by repeatedly applying division and remainder operations.

2. Step 2: Differentiate GCD from LCM

   While GCD finds the largest number dividing both inputs, LCM finds the smallest number divisible by both. The Euclidean algorithm is specifically for GCD.

3. Final Answer:

   The Greatest Common Divisor (GCD) -> Option C

4. Quick Check:

   Euclidean algorithm = GCD [OK]

Hint: Euclidean algorithm always finds GCD, not LCM [OK]

Common Mistakes:

• Confusing GCD with LCM
• Thinking it calculates sum or difference
• Assuming it finds prime factors

2. Which of the following TypeScript function signatures correctly defines a function to compute GCD using the Euclidean algorithm?

easy

A. function gcd(a: number, b: number): number

B. function gcd(a: string, b: string): string

C. function gcd(a: number): number

D. function gcd(): void

Solution

1. Step 1: Check parameter types for GCD function

   GCD requires two numbers as input, so parameters must be two numbers.

2. Step 2: Check return type

   The function returns a number representing the GCD, so return type must be number.

3. Final Answer:

   function gcd(a: number, b: number): number -> Option A

4. Quick Check:

   Two number inputs and number output [OK]

Hint: GCD needs two number inputs and returns a number [OK]

Common Mistakes:

• Using string types for numeric operations
• Missing one parameter
• Returning void instead of number

3. What is the output of the following TypeScript code?

function gcd(a: number, b: number): number {
  while (b !== 0) {
    const temp = b;
    b = a % b;
    a = temp;
  }
  return a;
}

console.log(gcd(48, 18));

medium

A. 12

B. 3

C. 18

D. 6

Solution

1. Step 1: Trace the Euclidean algorithm steps for gcd(48, 18)

   48 % 18 = 12, then gcd(18, 12); 18 % 12 = 6, then gcd(12, 6); 12 % 6 = 0, loop ends, gcd is 6.

2. Step 2: Confirm final returned value

   The function returns 6 after the loop ends.

3. Final Answer:

   6 -> Option D

4. Quick Check:

   GCD(48,18) = 6 [OK]

Hint: Use remainder steps until zero remainder [OK]

Common Mistakes:

• Returning the wrong variable after loop
• Confusing modulo operation
• Stopping too early in the loop

4. Identify the error in this TypeScript function for GCD and fix it:

function gcd(a: number, b: number): number {
  while (a !== 0) {
    const temp = a;
    a = b % a;
    b = temp;
  }
  return b;
}

medium

A. Replace '%' with '/' operator in the loop

B. No error, function works correctly

C. Change 'while (a !== 0)' to 'while (b !== 0)' and swap variables accordingly

D. Change 'return b;' to 'return a;' without other changes

Solution

1. Step 1: Test the function with example inputs

   Trace gcd(48, 18): starts a=48 b=18 -> a=18 b=48 -> a=12 b=18 -> a=6 b=12 -> a=0 b=6, returns 6 (correct GCD).

2. Step 2: Verify algorithm correctness

   This is a valid variant of the Euclidean algorithm with swapped roles (gcd(a,b) = gcd(b % a, a)). No syntax/runtime errors, handles edges like gcd(48,0)=48 correctly.

3. Final Answer:

   No error, function works correctly -> Option B

4. Quick Check:

   Equivalent Euclidean variant computes correct GCD [OK]

Hint: Non-standard loop condition but correct GCD computation [OK]

Common Mistakes:

• Assuming loop must check b !== 0
• Proposing to change return to a (breaks it)
• Suggesting / instead of %

5. Given two numbers 24 and 36, which of the following TypeScript code snippets correctly calculates their Least Common Multiple (LCM) using the GCD function?

hard

A. const lcm = (a: number, b: number): number => (a * b) / gcd(a, b);

B. const lcm = (a: number, b: number): number => gcd(a, b) / (a * b);

C. const lcm = (a: number, b: number): number => a + b - gcd(a, b);

D. const lcm = (a: number, b: number): number => Math.max(a, b);

Solution

1. Step 1: Recall the formula relating LCM and GCD

   LCM(a, b) = (a * b) / GCD(a, b). This formula uses the GCD to find the LCM efficiently.

2. Step 2: Check each code snippet against the formula

   const lcm = (a: number, b: number): number => (a * b) / gcd(a, b); matches the formula exactly. Options B, C, and D do not correctly calculate LCM.

3. Final Answer:

   const lcm = (a: number, b: number): number => (a * b) / gcd(a, b); -> Option A

4. Quick Check:

   LCM = (a*b)/GCD(a,b) [OK]

Hint: LCM = product divided by GCD [OK]

Common Mistakes:

• Dividing GCD by product instead of product by GCD
• Adding or subtracting instead of dividing
• Using max instead of LCM formula