Modular Arithmetic
Modular arithmetic studies the remainder when integers are divided, with applications in cryptography, computer science, and number theory.
What You Need to Know
Key Concept Diagram
a mod n is the remainder when a is divided by n
We say a is congruent to b modulo n (written a ≡ b mod n) when they have the same remainder
Modular arithmetic follows rules for addition, subtraction, and multiplication
Clock arithmetic is a real-world example: 11 + 3 = 2 (mod 12)
Fermat's Little Theorem and RSA encryption both rely on modular arithmetic
Key Vocabulary
Modulo
The remainder operation: a mod n gives the remainder when a is divided by n
Congruence
a ≡ b (mod n) means a and b have the same remainder when divided by n
Residue
The remainder obtained in modular arithmetic
Clock arithmetic
An everyday example of modular arithmetic using a 12-hour or 24-hour cycle
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is 17 mod 5?
Question 2
If it is currently 10 o'clock, what time will it be in 27 hours?
Question 3
Which statement is true? 25 ≡ ? (mod 7)
Key Concepts Summary
- ●a mod n is the remainder when a is divided by n
- ●We say a is congruent to b modulo n (written a ≡ b mod n) when they have the same remainder
- ●Modular arithmetic follows rules for addition, subtraction, and multiplication
- ●Clock arithmetic is a real-world example: 11 + 3 = 2 (mod 12)
- ●Fermat's Little Theorem and RSA encryption both rely on modular arithmetic