Fermat’s Little Theorem | Mathematics

Last Updated : 8 Jul, 2026

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 ap−1 is divided by p, the remainder is 1.

This also can be written as ap ≡ 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 = cd (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 an-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 ≡ ap-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).

➣Practice: Solved Examples

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