Modular Arithmetic Basics in DSA C# - Execution Trace

Concept Flow - Modular Arithmetic Basics

Start with two numbers a and b

↓

Choose modulus m

↓

Calculate a mod m

↓

Calculate b mod m

↓

Perform operation (add, sub, mul) on results

↓

Take result mod m

↓

Output result

Start with numbers a and b and modulus m, reduce a and b modulo m, perform arithmetic, then reduce the result modulo m.

Execution Sample

DSA C#

int a = 17;
int b = 23;
int m = 5;
int result = (a + b) % m;
Console.WriteLine(result);

Adds 17 and 23, then finds the remainder when divided by 5.

Execution Table

Step	Operation	Value(s) Involved	Intermediate Result	Result mod m	Output
1	Calculate a mod m	a=17, m=5	17 % 5 = 2	2
2	Calculate b mod m	b=23, m=5	23 % 5 = 3	3
3	Add results	2 + 3	5
4	Calculate (a+b) mod m	5 % 5	0	0
5	Print result	result=0			0

💡 Completed modular addition and printed the result.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	Final
a	17	17	17	17	17	17
b	23	23	23	23	23	23
m	5	5	5	5	5	5
a mod m	N/A	2	2	2	2	2
b mod m	N/A	N/A	3	3	3	3
sum	N/A	N/A	N/A	5	5	5
result	N/A	N/A	N/A	N/A	0	0

Key Moments - 3 Insights

Why do we take 'a mod m' and 'b mod m' before adding?

Taking 'a mod m' and 'b mod m' first keeps numbers small and prevents overflow, as shown in steps 1 and 2 of the execution_table.

Why do we take the final result mod m again after addition?

Because adding two numbers mod m can exceed m, we take mod m again to keep the result within the range 0 to m-1, as shown in step 4.

What happens if the sum equals m exactly?

If the sum equals m, then sum mod m is 0, as in step 4 where 5 % 5 = 0.

Visual Quiz - 3 Questions

Look at the execution_table, what is the value of 'a mod m' after step 1?

A 0

B 2

C 3

D 5

💡 Hint

Check the 'Intermediate Result' column in row for step 1.

At which step does the final modular result get calculated?

A Step 2

B Step 3

C Step 4

D Step 5

💡 Hint

Look for the step where '(a+b) mod m' is computed.

If m was changed to 6, what would be the output after step 5?

A 4

B 0

C 5

D 3

💡 Hint

Calculate (17 + 23) % 6 and check the final output.

Concept Snapshot

Modular Arithmetic Basics:
- Use '%' operator to find remainder.
- Reduce operands modulo m before operations.
- Perform arithmetic (add, sub, mul) on reduced values.
- Take modulo m of the result to keep within range.
- Result is always between 0 and m-1.

Full Transcript

Modular arithmetic means working with remainders after division by a number m. We start with two numbers a and b, and a modulus m. First, we find the remainder of a divided by m, called a mod m. Then we find b mod m. We add these two remainders. Since the sum might be bigger than m, we take the remainder of the sum divided by m again. This final result is the modular sum. For example, with a=17, b=23, and m=5, a mod m is 2, b mod m is 3, their sum is 5, and 5 mod 5 is 0. So the output is 0. This keeps numbers small and within the range 0 to m-1.

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