Fermat’s Little Theorem - Statement, Proof with Example

Fermat Little Theorem also known as Fermat 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

Let p be a prime number, and a be an integer such that a is not divisible by p. Consider the set of numbers formed by multiplying a with integers from 1 to p−1:

S = {a, 2a, 3a, . . . ,(p − 1)a}

Assume that for two numbers i and j such that 1 ≤ i, j < p, the products ia ≡ ja (mod p).

• If ia ≡ ja (mod p), then (i − j)a ≡ 0 (mod p).
• Since p is prime and a is not divisible by p, this implies that p must divide i − j.
• However, 1 ≤ i, j < p means that i − j cannot be divisible by p unless i = j.

Thus, all the products a, 2a, . . . ,(p − 1)a a are distinct modulo p.

The product of all the elements in S is:

a ⋅ 2a ⋅ 3a . . . (p − 1)a = a^p−1 ⋅ [1 ⋅ 2 ⋅ 3 ⋯ (p−1)] = a^p−1 ⋅ (p − 1)! . . . (i)

Note: 1 ⋅ 2 ⋅ 3 ⋅ . . . ⋅ (p−1) is the product of all integers from 1 to p − 1, which is known as (p − 1)! (the factorial of p − 1).

By Wilson's theorem, we know that:

(p−1)! ≡ −1 (mod p).

Thus, the equation (i) becomes:

(p−1)! ⋅ a^p−1 ≡ −1 ⋅ a^p−1 .

Since (p−1)! ≡ −1 (mod p), we rewrite the equation as:

−1 ⋅ a^p−1 ≡ −1 (mod p)

Multiplying both sides by −1, we get:

a^p−1 ≡ 1 (mod p)

Which is the required result.

Examples of Fermat's Little Theorem

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:

a^p−1 ≡ 1 (mod p)

where p is a prime number, and a is an integer not divisible by p.

Here, a = 7 and p = 13. By Fermat's Little Theorem:

7¹² ≡ 1 (mod 13)

7¹⁰⁰ = 7^12×8+4 = (7¹²)⁸ ⋅ 7⁴

By Fermat’s Little Theorem:

(7¹²)⁸ ≡ 1⁸ ≡ 1 (mod 13)

Therefore:

7¹⁰⁰ ≡ 7⁴ (mod 13)a^p−1 ≡ 1 (mod p)

Now, 7² = 49 ⇒ 49 ÷ 13 = 3 remainder 10

Thus: 7² ≡ 10 (mod 13)a^p−1 ≡ 1 (mod p)

Now: 7⁴ ≡ (7²)² ≡ 10² ≡ 100 (mod p)

Again, reducing 100 modulo 13:

100 ÷ 13 = 7 remainder 9

Thus: 7⁴ ≡ 9 (mod 13).

Result: Remainder when 7¹⁰⁰ 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.

Fermat's Little Theorem states:

a^p−1 ≡ 1 (mod p)

where p is a prime number and a is an integer not divisible by p.

Here, a = 3 and p = 53, so:

3⁵² ≡ 1 (mod 53)

3^100,000 = 3^52×1923+4 = (3⁵²)¹⁹²³ ⋅ 34

By Fermat's Little Theorem:

(3⁵²)¹⁹²³ ≡ 1¹⁹²³ ≡ 1 (mod 53)

Thus: 3^100,000 ≡ 3⁴(3⁵²)¹⁹²³ ⋅ 3⁴

3⁴ = 3 × 3 × 3 × 3 = 81

Now, reduce 81 modulo 53:

81÷53=1 remainder 28

Thus:

3⁴ ≡ 28 (mod 53)

Thus, 3^100,000 = 28 (mod 53)

Result: Remainder when 3^100,000 is divided by 53 is 28.

Worksheet on Fermat’s Little Theorem

You can download the free worksheet on Fermat’s Little Theorem with answer key from below:

Conclusion

Fermat’s Little Theorem is a powerful tool in number theory that helps us work with large numbers and modular arithmetic efficiently. It tells us that for any integer a that is not divisible by a prime number p, the expression a^p−1will leave a remainder of 1 when divided by p.

Read More,

• Prime Number
• Number Theory
• Composite Number
• Modular Arithmetic
• Divisibility Rules