Modular Arithmetic Basics in DSA Typescript - Build from Scratch

Modular Arithmetic Basics

📖 Scenario: Imagine you are working with a clock that shows numbers from 0 to 11. When the hour passes 11, it goes back to 0. This is an example of modular arithmetic.We will learn how to use modular arithmetic to find the remainder when numbers wrap around a fixed size, like hours on a clock.

🎯 Goal: You will create a small program that calculates the remainder of numbers when divided by a fixed number (called the modulus). This helps understand how numbers wrap around in modular arithmetic.

📋 What You'll Learn

Create a variable with a number to divide

Create a variable for the modulus (the number to divide by)

Calculate the remainder using modular arithmetic

Print the result showing the remainder

💡 Why This Matters

🌍 Real World

Modular arithmetic is used in clocks, calendars, computer hashing, and cryptography to handle numbers that wrap around fixed limits.

💼 Career

Understanding modular arithmetic is important for software developers working with algorithms, security, and systems programming.

1

Create the number to divide

Create a variable called number and set it to 29.

DSA Typescript

// Your code here

Hint

Use const number = 29; to create the variable.

2

Create the modulus value

Create a variable called modulus and set it to 12.

DSA Typescript

const number = 29;
// Your code here

Hint

Use const modulus = 12; to create the modulus variable.

3

Calculate the remainder using modular arithmetic

Create a variable called remainder and set it to the remainder of number divided by modulus using the % operator.

DSA Typescript

const number = 29;
const modulus = 12;
// Your code here

Hint

Use const remainder = number % modulus; to calculate the remainder.

4

Print the remainder result

Use console.log to print the text "Remainder: " followed by the value of remainder.

DSA Typescript

const number = 29;
const modulus = 12;
const remainder = number % modulus;
// Your code here

Hint

Use console.log("Remainder: " + remainder); to print the result.

Practice

1. What is the result of 7 % 3 in modular arithmetic?

easy

A. 2

B. 1

C. 3

D. 0

Solution

1. Step 1: Understand the modulo operation

   The modulo operator (%) gives the remainder when the first number is divided by the second.

2. Step 2: Calculate 7 % 3

   7 divided by 3 is 2 with a remainder of 1, but since 7 = 3*2 + 1, the remainder is 1.

3. Final Answer:

   1 -> Option A

4. Quick Check:

   7 % 3 = 1 [OK]

Hint: Modulo gives remainder after division [OK]

Common Mistakes:

• Confusing remainder with quotient
• Thinking modulo returns division result
• Mixing up dividend and divisor

2. Which of the following is the correct TypeScript syntax to compute the remainder of a divided by b?

easy

A. let result = a % b;

B. let result = a / b;

C. let result = a mod b;

D. let result = a & b;

Solution

1. Step 1: Recall TypeScript modulo operator

   In TypeScript, the modulo operator is represented by the percent sign (%).

2. Step 2: Identify correct syntax

   let result = a % b; uses a % b, which correctly computes the remainder.

3. Final Answer:

   let result = a % b; -> Option A

4. Quick Check:

   Modulo operator is % in TypeScript [OK]

Hint: Use % for modulo in TypeScript [OK]

Common Mistakes:

• Using / instead of %
• Using 'mod' keyword which doesn't exist
• Using bitwise & operator by mistake

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

const a = 14;
const b = 5;
console.log(a % b);

medium

A. 3

B. 4

C. 2

D. 0

Solution

1. Step 1: Calculate 14 % 5

   Divide 14 by 5: 5 * 2 = 10, remainder is 14 - 10 = 4.

2. Step 2: Confirm output

   The console.log prints the remainder, which is 4.

3. Final Answer:

   4 -> Option B

4. Quick Check:

   14 % 5 = 4 [OK]

Hint: Modulo is remainder after division [OK]

Common Mistakes:

• Calculating quotient instead of remainder
• Mixing up operands order
• Assuming modulo returns division result

4. Identify the error in this TypeScript code snippet that tries to compute modulo:

function mod(a: number, b: number): number {
  return a % b;
}

console.log(mod(10, 0));

medium

A. Syntax error due to missing semicolon

B. No error, outputs 0

C. Runtime error: division by zero

D. Returns NaN without error

Solution

1. Step 1: Analyze modulo with zero divisor

   In TypeScript/JavaScript, modulo by zero returns NaN and does NOT throw a runtime error.

2. Step 2: Understand error behavior

   Calling mod(10, 0) computes 10 % 0 which is NaN, and console.log prints NaN.

3. Final Answer:

   Returns NaN without error -> Option D

4. Quick Check:

   10 % 0 = NaN [OK]

Hint: Modulo by zero returns NaN in TypeScript [OK]

Common Mistakes:

• Thinking modulo by zero causes runtime error
• Assuming modulo by zero returns 0
• Confusing with languages that raise exceptions

5. You want to find the remainder when the sum of two large numbers a = 123456789 and b = 987654321 is divided by m = 1000. Which expression correctly uses modular arithmetic to compute (a + b) % m without overflow?

hard

A. (a + b) % m

B. a % (b % m)

C. (a % m + b % m) % m

D. (a % b) + (b % m)

Solution

1. Step 1: Understand modular addition property

   Modular arithmetic allows splitting sums: (a + b) % m = ((a % m) + (b % m)) % m.

2. Step 2: Apply property to avoid overflow

   Computing (a % m + b % m) % m prevents large intermediate sums and overflow.

3. Step 3: Check other options

   (a + b) % m risks overflow; B and D misuse modulo operations.

4. Final Answer:

   (a % m + b % m) % m -> Option C

5. Quick Check:

   Use modular addition property [OK]

Hint: Add remainders then modulo again to avoid overflow [OK]

Common Mistakes:

• Directly adding large numbers before modulo
• Misusing modulo on operands incorrectly
• Ignoring overflow in large sums