Modular Arithmetic Basics in DSA C# - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Modular Addition

What is the output of the following C# code that performs modular addition?

DSA C#

int a = 17;
int b = 23;
int mod = 10;
int result = (a + b) % mod;
Console.WriteLine(result);

A 0

B 10

C 1

D 5

💡 Hint

Remember to add first, then take the remainder with the modulus.

The sum is 17 + 23 = 40. Taking 40 % 10 gives 0, so the correct output is 0.

❓ Predict Output

intermediate

Modular Multiplication Result

What will be printed after running this C# code snippet that uses modular multiplication?

DSA C#

int x = 7;
int y = 8;
int mod = 12;
int result = (x * y) % mod;
Console.WriteLine(result);

A 4

B 8

C 10

D 0

💡 Hint

Multiply first, then apply the modulus operator.

7 * 8 = 56. Then 56 % 12 = 8, so output should be 8.

❓ Predict Output

advanced

Modular Exponentiation Output

What is the output of this C# code that calculates modular exponentiation?

DSA C#

int baseNum = 3;
int exponent = 4;
int mod = 5;
int result = 1;
for (int i = 0; i < exponent; i++)
{
    result = (result * baseNum) % mod;
}
Console.WriteLine(result);

A 3

B 4

C 1

D 2

💡 Hint

Calculate 3^4 mod 5 step by step.

3^4 = 81. 81 % 5 = 1, but let's check the loop calculation carefully.

Step 1: result=1*3=3 %5=3

Step 2: 3*3=9 %5=4

Step 3: 4*3=12 %5=2

Step 4: 2*3=6 %5=1

Final result is 1.

❓ Predict Output

advanced

Modular Inverse Calculation

What is the output of this C# code that tries to find the modular inverse of 3 modulo 11 using brute force?

DSA C#

int a = 3;
int mod = 11;
int inverse = -1;
for (int x = 1; x < mod; x++)
{
    if ((a * x) % mod == 1)
    {
        inverse = x;
        break;
    }
}
Console.WriteLine(inverse);

A 4

B 8

C 7

D -1

💡 Hint

Find x such that (3 * x) % 11 == 1.

Check multiples of 3 mod 11: 3*7=21 %11=10, no; 3*4=12 %11=1, yes.

Actually, 3*4=12 %11=1, so inverse is 4.

🧠 Conceptual

expert

Properties of Modular Arithmetic

Which of the following statements about modular arithmetic is FALSE?

A If a ≡ b (mod m), then (a * c) ≡ (b * c) (mod m) for any integer c.

B If a ≡ b (mod m), then (a + c) ≡ (b + c) (mod m) for any integer c.

C If a ≡ b (mod m) and c ≡ d (mod m), then (a - c) ≡ (b - d) (mod m).

D If a ≡ b (mod m), then a^c ≡ b^c (mod m) for any integer c, including negative c.

💡 Hint

Consider the meaning of negative exponents in modular arithmetic.

Modular arithmetic properties hold for addition, subtraction, and multiplication.

However, exponentiation with negative exponents requires modular inverse, which may not exist.

So statement A is false.

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