What Is A Modulus Of A Number? Complete Guide with Examples

Quick Answer: What Is a Modulus of a Number?

The modulus of a number is the remainder left over after dividing it by another number. Written as a mod n, it always gives a result between 0 and n−1. For example, 17 mod 5 = 2, because 17 ÷ 5 leaves a remainder of 2. In programming, the modulus operator is the percent sign (%), and it is used constantly for tasks like checking even/odd numbers, wrapping array indexes, and clock arithmetic.

Quick Summary: Modulus of a Number

• Modulus finds the remainder after division (e.g., 17 mod 5 = 2)
• Formula: a mod n = remainder when dividing a by n
• Symbol: % in programming, « mod » in math
• Applications: Even/odd checks, clock arithmetic, cryptography, programming
• Range: Result is always between 0 and (n-1)

💡 Quick Example

17 mod 5 = ?

17 ÷ 5 = 3 with a remainder of 2

Therefore: 17 mod 5 = 2

Division Breakdown Example

Let’s divide 17 by 5:

• 17 ÷ 5 = 3 remainder 2
• This means: 17 = (5 × 3) + 2
• The quotient is 3
• The remainder is 2
• Therefore: 17 mod 5 = 2

There is a small but very important distinction to keep in mind:

When you divide 17 by 5:

• The Quotient (result of division) is 3 → This tells you how many full groups of 5 you can make
• The Modulus (result of mod operation) is 2 → This tells you only what is left over after making those groups

Visual Example: 17 Items in Groups of 5

How to Answer Different Questions

If someone asks « What is 17 ÷ 5? », the answer is: 3 remainder 2

But if someone asks « What is 17 mod 5? », they only want the remainder: 2

Summary Table

Modulus Formula

a mod n = r

Where:

• a = dividend (number being divided)
• n = divisor (number dividing by)
• r = remainder (result of modulus operation)

Important: The remainder r must always satisfy: 0 ≤ r < n

1. Divide: 10 ÷ 3 = 3 remainder 1
2. Verify: 3 × 3 = 9, and 10 – 9 = 1
3. Result: 10 mod 3 = 1

1. Divide: 25 ÷ 7 = 3 remainder 4
2. Verify: 7 × 3 = 21, and 25 – 21 = 4
3. Result: 25 mod 7 = 4

1. Divide: 20 ÷ 5 = 4 remainder 0
2. Verify: 5 × 4 = 20, and 20 – 20 = 0
3. Result: 20 mod 5 = 0
4. Note: When remainder is 0, the dividend is evenly divisible by the divisor

1. Divide: 7 ÷ 10 = 0 remainder 7
2. Verify: 10 × 0 = 0, and 7 – 0 = 7
3. Result: 7 mod 10 = 7
4. Note: When dividend < divisor, the modulus equals the dividend

1. Range of Results

For a mod n, the result is always: 0 ≤ r < n

Example: Any number mod 5 will give: 0, 1, 2, 3, or 4

2. Distributive Property (Addition)

(a + b) mod n = [(a mod n) + (b mod n)] mod n

Example:

(13 + 17) mod 5 = [(13 mod 5) + (17 mod 5)] mod 5

30 mod 5 = [3 + 2] mod 5 = 5 mod 5 = 0

3. Distributive Property (Multiplication)

(a × b) mod n = [(a mod n) × (b mod n)] mod n

Example:

(8 × 6) mod 5 = [(8 mod 5) × (6 mod 5)] mod 5

48 mod 5 = [3 × 1] mod 5 = 3

4. Zero Modulus Special Case

a mod n = 0 means a is evenly divisible by n

Example: 15 mod 5 = 0 (because 15 = 5 × 3, no remainder)

5. Identity Property

If a < n, then a mod n = a

Example: 3 mod 10 = 3 (because 3 is smaller than 10)

Examples:

• 10 mod 2 = 0 → 10 is even
• 15 mod 2 = 1 → 15 is odd
• 144 mod 2 = 0 → 144 is even

Problem: What time is it 25 hours from now if it’s currently 10:00?

Solution:

• Current time: 10:00
• Add 25 hours: 10 + 25 = 35
• Apply mod 12: 35 mod 12 = 11
• Answer: 11:00

// Example: Array with 5 elements
// When at index 4, next is (4+1) % 5 = 0 (wraps to start)

Example: Is 144 divisible by 12?

144 mod 12 = 0 → Yes, 144 is evenly divisible by 12

# Check if number is even
if num % 2 == 0:
    print("Even")

// Check if number is odd
if (num % 2 === 1) {
    console.log("Odd");
}

// Check divisibility
if (num % 3 == 0) {
    System.out.println("Divisible by 3");
}

1. Calculate: 14 mod 3

2. Calculate: 20 mod 6

3. Is 24 even or odd? (Use modulus)

4. Calculate: 100 mod 10

5. Calculate: 5 mod 8

6. What time is it 30 hours after 8:00? (Use mod 12)

7. Is 157 divisible by 5? (Use modulus)

8. Calculate: 99 mod 7

9. Find all possible results of: n mod 4

10. If today is Wednesday (day 3), what day is it 20 days from now? (Use mod 7)

11. Calculate: (15 + 23) mod 8

12. Calculate: (6 × 7) mod 5

13. What is the last digit of 7^100? (Hint: Use modulus)

14. In an array of length 10, if current index is 8 and we move forward 5 positions, what’s the new index?

15. Is 2,468 divisible by both 2 and 4?

Easy Level Solutions

1. 14 mod 3 = 2 (14 ÷ 3 = 4 remainder 2)

2. 20 mod 6 = 2 (20 ÷ 6 = 3 remainder 2)

3. 24 mod 2 = 0, so 24 is even

4. 100 mod 10 = 0

5. 5 mod 8 = 5 (dividend < divisor)

Medium Level Solutions

6. (8 + 30) mod 12 = 38 mod 12 = 2 → 2:00

7. 157 mod 5 = 2 (not 0), so not divisible by 5

8. 99 mod 7 = 1 (99 ÷ 7 = 14 remainder 1)

9. Possible results: 0, 1, 2, 3

10. (3 + 20) mod 7 = 23 mod 7 = 2 → Tuesday (day 2)

Challenge Level Solutions

11. (15 + 23) mod 8 = 38 mod 8 = 6

12. (6 × 7) mod 5 = 42 mod 5 = 2

13. Look for pattern: 7^1=7, 7^2=49 (last digit 9), 7^3=343 (last digit 3), 7^4=2401 (last digit 1). Pattern repeats every 4. 100÷4=25, so last digit is 1

14. (8 + 5) mod 10 = 13 mod 10 = 3

15. 2468 mod 2 = 0 (even), 2468 mod 4 = 0. Yes, divisible by both

Q1: What is modulus in math?

A: Modulus is a mathematical operation that finds the remainder after dividing one number by another. For example, 17 mod 5 = 2 because 17 ÷ 5 = 3 with remainder 2.

Q2: How do you calculate modulus?

A: To calculate a mod n: divide a by n, then take the remainder. For example, 25 mod 7: divide 25 by 7 to get 3 remainder 4, so 25 mod 7 = 4.

Q3: What is the modulus symbol?

A: In programming, modulus is represented by the percent symbol (%). In mathematics, it is written as "mod", for example: 17 mod 5.

Q4: What is modulus used for?

A: Modulus is used for checking even/odd numbers, clock arithmetic, array indexing, cryptography, hash functions, and determining divisibility.

Q5: Can modulus be negative?

A: In most programming languages, modulus with negative numbers is handled differently. In mathematics, the result is typically defined to be non-negative (0 ≤ r < n).

Q6: What does 0 mod n equal?

A: 0 mod n always equals 0 for any positive n, because 0 divided by any number gives quotient 0 and remainder 0.

Q7: Is modulus the same as remainder?

A: Yes! Modulus and remainder mean the same thing. Both refer to what is left over after division.

Q8: What is the difference between division and modulus?

A: Division gives you the quotient (how many times the divisor goes into the dividend), while modulus gives you the remainder (what is left over).

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