Modular Arithmetic Basics in DSA Javascript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is modular arithmetic?

Modular arithmetic is a system of arithmetic for integers where numbers wrap around after reaching a certain value called the modulus.

beginner

What does the expression 'a % n' mean in modular arithmetic?

It means the remainder when integer 'a' is divided by integer 'n'. This remainder is always between 0 and n-1.

beginner

How do you add two numbers modulo n? (a + b) % n

Add the two numbers normally, then take the remainder when divided by n. This keeps the result within 0 to n-1.

intermediate

Why is modular arithmetic useful in programming?

It helps keep numbers within a fixed range, prevents overflow, and is used in hashing, cryptography, and cyclic structures.

beginner

What is the result of (17 % 5) and why?

The result is 2 because 17 divided by 5 is 3 with a remainder of 2.

What is the value of 10 % 3?

A 2

B 3

C 1

D 0

10 divided by 3 is 3 with remainder 1, so 10 % 3 = 1.

Which of these is true for modular addition (a + b) % n?

A Result is always less than n

B Result can be greater than n

C Result is always equal to a + b

D Result is always zero

Modular addition wraps the sum back into the range 0 to n-1, so result is always less than n.

If n = 7, what is (15 % n)?

A 2

B 1

C 0

D 7

15 divided by 7 is 2 with remainder 1, so 15 % 7 = 1.

Why do programmers use modular arithmetic?

A To increase number size

B To sort arrays

C To avoid division

D To keep numbers within a fixed range

Modular arithmetic keeps numbers within a fixed range, useful for preventing overflow and cyclic operations.

What is the remainder when 23 is divided by 5?

A 3

B 4

C 2

D 1

23 divided by 5 is 4 with remainder 3, so 23 % 5 = 3.

Explain modular arithmetic and how the modulus operator works with examples.

Think about how numbers wrap around after reaching the modulus.

Describe why modular arithmetic is important in programming and give two use cases.

Consider how modular arithmetic helps with fixed ranges and cycles.

Practice

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

easy

A. 2

B. 3

C. 1

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 7 divided by 3

   7 divided by 3 is 2 with a remainder of 1.

3. Final Answer:

   1 -> Option C

4. Quick Check:

   7 % 3 = 1 [OK]

Hint: Modulus gives remainder after division [OK]

Common Mistakes:

• Confusing modulus with division
• Using 3 instead of remainder
• Thinking modulus returns quotient

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

easy

A. a % b

B. a mod b

C. mod(a, b)

D. a // b

Solution

1. Step 1: Recall JavaScript modulus syntax

   In JavaScript, the modulus operator is the percent sign (%) placed between two numbers or variables.

2. Step 2: Identify correct syntax

   Only a % b correctly computes the remainder of a divided by b.

3. Final Answer:

   a % b -> Option A

4. Quick Check:

   Use % for modulus in JavaScript [OK]

Hint: Use % operator for modulus in JavaScript [OK]

Common Mistakes:

• Using 'mod' keyword
• Using integer division operator //
• Calling a non-existent mod() function

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

const x = 15;
const y = 4;
console.log(x % y);

medium

A. 4

B. 3

C. 0

D. 1

Solution

1. Step 1: Calculate 15 divided by 4

   15 divided by 4 is 3 with a remainder of 3.

2. Step 2: Apply modulus operator

   The expression 15 % 4 returns the remainder, which is 3.

3. Final Answer:

   3 -> Option B

4. Quick Check:

   15 % 4 = 3 [OK]

Hint: Modulus returns remainder after division [OK]

Common Mistakes:

• Confusing remainder with quotient
• Expecting 4 instead of remainder
• Mistaking modulus for division

4. Find the error in the following code snippet that tries to compute the modulus of two numbers:

let a = 10;
let b = 3;
let result = a % b();
console.log(result);

medium

A. Missing semicolon after console.log

B. Modulus operator should be replaced with division

C. Variables a and b should be strings

D. Using parentheses after b causes an error

Solution

1. Step 1: Analyze the modulus expression

   The code uses a % b(), which treats b as a function call, but b is a number, not a function.

2. Step 2: Identify the error

   Parentheses after b cause a runtime error because numbers cannot be called like functions.

3. Final Answer:

   Using parentheses after b causes an error -> Option D

4. Quick Check:

   Variables are not functions, so no () after b [OK]

Hint: Don't add () after variables unless they are functions [OK]

Common Mistakes:

• Adding () after number variables
• Confusing modulus with division
• Ignoring runtime errors

5. You want to find the remainder when the sum of two numbers a = 123456789 and b = 987654321 is divided by 1000. Which expression correctly computes this using modular arithmetic properties?

hard

A. ((a % 1000) + (b % 1000)) % 1000

B. (a % 1000) + (b % 1000)

C. a + b % 1000

D. a % (b % 1000)

Solution

1. Step 1: Recall modular addition property

   The remainder of a sum modulo m equals the sum of remainders modulo m: (a + b) % m = ((a % m) + (b % m)) % m.

2. Step 2: Apply property to given numbers

   Compute (a % 1000) and (b % 1000), add them, then take modulo 1000 again to get correct remainder.

3. Final Answer:

   ((a % 1000) + (b % 1000)) % 1000 -> Option A

4. Quick Check:

   Use modular addition: ((a%m)+(b%m))%m [OK]

Hint: Add remainders then mod again for sum modulus [OK]

Common Mistakes:

• Skipping final modulus after addition
• Adding without modulus
• Using wrong modulus order