Modular Arithmetic Basics in DSA C# - Build from Scratch

Modular Arithmetic Basics

📖 Scenario: Imagine you are working with a clock that only shows hours from 0 to 11. When you add hours, the clock resets after 11 back to 0. This is an example of modular arithmetic.

🎯 Goal: You will create a simple program to calculate the result of adding hours on a 12-hour clock using modular arithmetic.

📋 What You'll Learn

Create an integer variable called currentHour with the value 9

Create an integer variable called hoursToAdd with the value 5

Calculate the new hour using modular arithmetic and store it in an integer variable called newHour

Print the value of newHour

💡 Why This Matters

🌍 Real World

Modular arithmetic is used in clocks, calendars, cryptography, and computer science to handle repeating cycles.

💼 Career

Understanding modular arithmetic helps in programming tasks involving cycles, hashing, and algorithms that require wrap-around logic.

1

Create the initial hour variable

Create an integer variable called currentHour and set it to 9.

DSA C#

# Your code here

Hint

Use int currentHour = 9; to create the variable.

2

Add hours to the current hour

Create an integer variable called hoursToAdd and set it to 5.

DSA C#

int currentHour = 9;
# Your code here

Hint

Use int hoursToAdd = 5; to create the variable.

3

Calculate the new hour using modular arithmetic

Create an integer variable called newHour and set it to the result of (currentHour + hoursToAdd) % 12.

DSA C#

int currentHour = 9;
int hoursToAdd = 5;
# Your code here

Hint

Use the modulus operator % to wrap around after 11.

4

Print the new hour

Write Console.WriteLine(newHour); to print the value of newHour.

DSA C#

int currentHour = 9;
int hoursToAdd = 5;
int newHour = (currentHour + hoursToAdd) % 12;
# Your code here

Hint

The output should be 2 because 9 + 5 = 14, and 14 % 12 = 2.

Practice

1. What is the result of 17 % 5 in modular arithmetic?

easy

A. 2

B. 3

C. 5

D. 0

Solution

1. Step 1: Understand the modulus operation

   The modulus operator (%) gives the remainder after division of the first number by the second.

2. Step 2: Calculate 17 % 5

   Dividing 17 by 5 gives 3 with a remainder of 2.

3. Final Answer:

   2 -> Option A

4. Quick Check:

   17 % 5 = 2 [OK]

Hint: Modulus gives remainder after division [OK]

Common Mistakes:

• Confusing modulus with division result
• Using quotient instead of remainder
• Thinking modulus returns zero if not divisible

2. Which of the following is the correct syntax to compute the modulus of two integers a and b in C#?

easy

A. int result = a & b;

B. int result = a mod b;

C. int result = a / b;

D. int result = a % b;

Solution

1. Step 1: Recall C# modulus operator

   In C#, the modulus operator is the percent sign (%).

2. Step 2: Check each option

   int result = a % b; uses % correctly. int result = a mod b; uses 'mod' which is not valid syntax. int result = a / b; uses division (/), not modulus. int result = a & b; uses bitwise AND (&), unrelated to modulus.

3. Final Answer:

   int result = a % b; -> Option D

4. Quick Check:

   C# modulus syntax = a % b [OK]

Hint: Use % symbol for modulus in C# [OK]

Common Mistakes:

• Using 'mod' keyword instead of %
• Confusing division (/) with modulus (%)
• Using bitwise operators (&) mistakenly

3. What is the output of the following C# code?

int a = 14;
int b = 4;
int c = (a + b) % 5;
Console.WriteLine(c);

medium

A. 2

B. 4

C. 3

D. 1

Solution

1. Step 1: Calculate sum inside parentheses

   a + b = 14 + 4 = 18.

2. Step 2: Compute modulus 18 % 5

   18 divided by 5 is 3 remainder 3, so 18 % 5 = 3.

3. Final Answer:

   3 -> Option C

4. Quick Check:

   (14 + 4) % 5 = 3 [OK]

Hint: Add first, then apply modulus [OK]

Common Mistakes:

• Applying modulus before addition
• Confusing remainder with quotient
• Incorrect arithmetic calculation

4. Identify the error in the following C# code snippet that tries to compute modulus:

int x = 10;
int y = 3;
int result = x % y;

medium

A. No error, code is correct

B. Missing semicolon after the first line

C. Using % operator with integers is invalid

D. Variables x and y must be declared as double

Solution

1. Step 1: Check syntax correctness

   Each line ends with a semicolon; variables are declared as integers.

2. Step 2: Validate modulus usage

   The modulus operator (%) works correctly with integers in C#.

3. Final Answer:

   No error, code is correct -> Option A

4. Quick Check:

   Modulus with int variables is valid [OK]

Hint: Modulus works with int types and needs semicolons [OK]

Common Mistakes:

• Thinking % is invalid for integers
• Missing semicolons (not in this code)
• Using floating types unnecessarily

5. You want to find the remainder when the sum of all numbers from 1 to 100 is divided by 7. Which C# expression correctly computes this using modular arithmetic?

hard

A. int result = 100 * (101 % 7) / 2;

B. int result = (100 * 101 / 2) % 7;

C. int result = (100 % 7) * (101 % 7) / 2;

D. int result = (100 + 101) % 7 / 2;

Solution

1. Step 1: Recall formula for sum 1 to n

   The sum of numbers from 1 to 100 is n(n+1)/2 = 100*101/2.

2. Step 2: Apply modulus after full sum calculation

   Compute (100*101/2) % 7 to get the remainder after division by 7.

3. Final Answer:

   int result = (100 * 101 / 2) % 7; -> Option B

4. Quick Check:

   Sum formula then modulus = correct [OK]

Hint: Calculate full sum first, then apply modulus [OK]

Common Mistakes:

• Applying modulus before multiplication or division
• Incorrect order of operations
• Dividing before multiplying causing wrong result