Euler's Theorem

Euler's Theorem is a key concept in number theory, named after the Swiss mathematician Leonhard Euler. It states that if a is any integer that is coprime with a positive integer m, then:

This means that raising a to the power of ϕ(m) (Euler’s Totient Function) will always leave a remainder of 1 when divided by m.
Interestingly, Fermat’s Little Theorem is just a special case of Euler’s Theorem. When m is a prime number p, ϕ(p) = p - 1, so Euler’s Theorem becomes Fermat’s Theorem.

Euler’s Theorem connects modular arithmetic and number theory, and it's especially useful in solving problems with large powers and in fields like cryptography.

Euler's Theorem Formula

Statement of Euler's Theorem: if a and n are coprime positive integers, then it can be used as a formula for further calculations, i.e.,

a^ϕ(n) ≡ 1 (mod n)

Where,

• a: Any integer
• n: A positive integer
• mod n represents congruence modulo n
• ≡ denotes equivalence,
• ϕ(n): Euler’s totient function, which counts how many numbers less than n are coprime to n

Proof of Euler's Theorem

Let φ(n) = k, and let {a₁, . . . , a_k} be a reduced residue system mod n.

For some a_i in {a₁, . . . , a_k}

Since (a, n) = 1, {aa₁, . . . , aa_k} is another reduced residue system mod n.

Since this is the same set of numbers mod n as the original system, the two systems must have the same product mod n:

(aa₁)· · ·(aa_k) = a₁ · · · a_k (mod n)

⇒ a^k (a₁ · · · a_k) = a₁ · · · ak (mod n)

Now each a_i is invertible mod n, so multiplying both sides by a₁⁻¹ · · · a_k⁻¹ , We get

a^k = 1 (mod n)

or a^φ(n) = 1 (mod n).

Example Showing Euler's Theorem Formula

Problem: Verify Euler's Theorem for a = 3 and n = 8.

Solution:

First, we calculate ϕ(8). The numbers less than 8 that are coprime to 8 are 1, 3, 5, and 7. Thus, ϕ(8) = 4.

Next, calculate 3⁴ and find its remainder when divided by 8

3⁴ = 81

Now, find 81 mod 8

81 mod 8 ≡ 1

Thus, 3⁴ ≡ 1 (mod 8), which verifies Euler's Theorem.

Euler's Totient Function

Formally, for a positive integer n, ϕ(n) is defined as follows:

ϕ(n) = count of integers 1 ≤ a <n such that gcd (a,n) = 1

Where:

• GCD(a, n) denotes the greatest common divisor of a and n.
• ϕ(n) represents the totient function of n.

Fermat's vs Euler's Theorems

Euler's Theorem is a generalization of Fermat's Theorem. Here are the key differences between Fermat's and Euler's Theorems:

Applications of Euler's Theorem

Euler's Theorem has many applications in a wide range of areas, such as mathematics and even elsewhere. Here are some notable applications:

RSA Encryption: Euler's theorem is foundational in modern cryptography, particularly in the RSA encryption algorithm. RSA utilizes Euler's theorem in the process of encryption and decryption. In RSA, the public and private keys are generated in such a way that they are inverses of each other modulo φ(n), where n is the product of two large prime numbers.

Problem Solving in Number Theory: Euler's theorem is a powerful tool in solving number theory problems involving divisibility, remainders, and the properties of numbers in different number systems.

Primality Testing: Euler's theorem is used in primality testing algorithms, such as the Fermat primality test. While this test is not infallible (it can give false positives for Carmichael numbers), it offers a quick way to check for non-prime numbers. If for some a coprime with φ(n) ≡ 1 (mod n), then n is not prime.

Mathematical Proofs: Euler's theorem is a general case for proofs enabling modular arithmetic, divisibility tests, and number theory identities, and it provides clear and convincing mathematical arguments that are the foundation of rigorous mathematical analysis.

Also Check:

• Extended Euclid Division Algorithm
• Euler's Formula

Solved Questions on Euler’s Theorem

Question 1: Find the remainder when 3¹⁰⁰ is divided by 7.

Solution:

Since 7 is a prime number, ϕ(7) = 7−1 = 6.

According to Euler's Theorem,
3⁶ ≡ 1 (mod 7).

Now, 3¹⁰⁰ can be rewritten as 3^6×16+4.

Modular exponentiation:
3¹⁰⁰ ≡ (3⁶)¹⁶ × 3⁴ ≡ 1¹⁶ × 81 ≡ 4 (mod 7).

So, when 3¹⁰⁰ is divided by 7, the remainder is 4.

Question 2: Find the remainder when 7²⁰ is divided by 21.

Solution:

Since 21 can be factored into 3 × 7, the GCD of 7 and 21 is 7, which is not 1, they are not coprime.

We have ϕ(21) = ϕ(3) ϕ(7)
ϕ(3) = 3 - 1 = 2
ϕ(7) = 7 - 1 = 6
ϕ(21) = ϕ(3) ϕ(7) = 2×6 = 12

According to Euler's Theorem,
7¹² ≡ 7 (mod 21).

Now, 7²⁰ can be expressed as 7^12×1+8.

Modular exponentiation:
7²⁰ ≡ (7¹²)¹ × 7⁸ ≡ 1 × 7⁸ ≡ 7 (mod 21)
7⁸ = (7⁴)² ≡ 7² ≡ 7 mod 21

Thus,
7²⁰ ≡ 7 mod 21

So, when 7²⁰ is divided by 21, the remainder is 7.

Practice Questions on Euler's Theorem

Question 1: Find the remainder when 2⁵⁰ is divided by 11.

Question 2: Calculate the remainder when 5¹⁰⁰ is divided by 17.

Question 3: Determine the remainder when 3⁷⁵ is divided by 13.

Question 4: Find the remainder when 4⁴⁰ is divided by 9.

Question 5: Find the remainder when 10²⁵ is divided by 8.

Working of Euler's Theorem

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