Fermat’s Little Theorem - Statement, Proof with Example

Fermat's Little Theorem, also known as Fermat's remainder theorem, is a fundamental result in number theory that deals with properties of prime numbers and modular arithmetic.

Fermat’s Little Theorem states that

"if p is a prime number and a is an integer such that a is not divisible by p, then a^{p-1} \equiv 1 \pmod{p}."

This means that when a^p−1 is divided by p, the remainder is 1.

This also can be written as a^p ≡ a (mod p).

For Example,

Let's take p = 7 (a prime number), and a = 3. According to the Fermat's Little Theorem :

3^{7-1} = 3^6 \equiv 1 \pmod{7}

This means when you calculate 3⁶ and divide it by 7, the remainder is 1.

Proof of Fermat's Little Theorem

Using Euler’s Theorem (Simplified Approach)

Fermat’s Little Theorem is a special case of Euler’s Theorem, which states:

If a is coprime to n, then:

a^{\phi(n)} \equiv 1 \ (\text{mod} \ n)

Where \phi(n) is Euler’s totient function (count of numbers less than n and coprime to it).

For a prime number p, \phi(p) = p - 1. So:

a^{p-1} \equiv 1 \ (\text{mod} \ p)

This completes the proof of Fermat’s Little Theorem in a concise and clear manner, without needing Wilson’s Theorem.

Using Wilson’s Theorem (Advanced)

If you want to explore a more involved proof, it goes like this:

Let: S = \{a, 2a, 3a, \ldots, (p - 1)a\}

Modulo p, this set contains p−1 distinct values (none repeat), because if:

ia \equiv ja \ (\text{mod} \ p) \Rightarrow (i - j)a \equiv 0 \Rightarrow i = j \ (\text{since } a \not\equiv 0 \ \text{and } p \ \text{is prime})

So: a \cdot 2a \cdot \ldots \cdot (p - 1)a = a^{p-1} \cdot (p - 1)!

By Wilson’s Theorem: (p - 1)! \equiv -1 \ (\text{mod} \ p)

So: a^{p-1} \cdot (p - 1)! \equiv (p - 1)! \ (\text{mod} \ p)

Divide both sides by (p - 1)! (valid since it’s not divisible by p):

a^{p-1} \equiv 1 \ (\text{mod} \ p)

Applications in Computer Science

Cryptography

• For RSA Decryption, to compute m = c^d (mod n) via Chinese Remainder Theorem
• In Diffie Hellmann Key Exchange Algorithm for secure key exchange by optimizing calculations of large powers modulo a prime.
• In Digital signature, ensures that inverse exists.

Primality Testing

• For an n to be prime, checking if a^n-1 = 1 (mod n). Exceptions for Carmichael numbers like 561, which is not a prime but still passes the test. This is the basis for Monte Carlo primality tests.

Algorithmic Optimizations

• For modular inverse computation, for prime p, a^-1 ≡ a^p-2 (mod p) and for exponent reduction; FLT allows reducing exponents, speeding up computations.

Hash Functions

• FLT ensures uniform distribution in hash functions using prime-sized tables (e.g., universal hashing).

Solved Examples

Example 1: Find the remainder when 7¹⁰⁰ is divided by 13.

Solution:

Since 13 is a prime number, we can apply Fermat's Little Theorem, which states:

7^{12} \equiv 1 \ (\text{mod} \ 13)

Now, 100= 12 ×8 +4 , so:

7^{100} = (7^{12})^8 \cdot 7^4 \equiv 1^8 \cdot 7^4 \equiv 7^4 \ (\text{mod} \ 13)

Calculate 7^4 \mod 13:

• 7^2 = 49 \Rightarrow 49 \mod 13 = 10
• 7^4 = (7^2)^2 = 10^2 = 100 \Rightarrow 100 \mod 13 = 9

Result: Remainder when 7^{100} is divided by 13 is 9.

Example 2: Find the remainder of 3^100,000 when divided by 53.

Solution:

Since 53 is a prime number, we can apply Fermat's Little Theorem.

3^{52} \equiv 1 \ (\text{mod} \ 53)

Now:

100000 = 52 \times 1923 + 4 \Rightarrow 3^{100000} = (3^{52})^{1923} \cdot 3^4 \equiv 1^{1923} \cdot 3^4 = 3^4 \ (\text{mod} \ 53)

• 3^4 = 81 \Rightarrow 81 \mod 53 = 28

Result: Remainder when 3^{100000} is divided by 53 is 28.

Practice Questions

1. Find the remainder when 5¹²³ is divided by 13.

2. Compute the remainder when 11⁵⁰ is divided by 17.

3. What is the remainder when 7³⁰⁰ is divided by 19?

4. Calculate the remainder of 2²⁰⁰ modulo 31.

5. Find the remainder when 8⁷⁵ is divided by 29.

6. Determine the remainder when 9¹⁰⁰ is divided by 23.

7. What is the remainder when 4²⁰²⁴ is divided by 7?

8. Compute the remainder of 6⁹⁹ modulo 37.

9. Find the remainder when 10⁵⁰⁰ is divided by 43.

10. What is the remainder when 3⁶⁴ is divided by 41?

Answer Keys

1. 8
2. 2
3. 1
4. 1
5. 2
6. 9
7. 2
8. 31
9. 9
10. 1

Related Articles:

• Program for Fermet's Little Theorem
• Prime Number
• Number Theory
• Composite Number
• Modular Arithmetic
• Divisibility Rules