Euler Totient Function Calculator With Steps

Calculator Form

Example Data Table

Formula Used

The Euler totient function is written as φ(n). It counts integers from 1 to n that are coprime with n.

Prime factor formula:

φ(n) = n × ∏(1 - 1/p)

Here, p means every distinct prime factor of n. Repeated powers are shown in factorization, but each prime appears once in the product formula.

Prime power rule: If n = p^k, then φ(n) = p^k - p^(k - 1).

Prime number rule: If n is prime, then φ(n) = n - 1.

How to Use This Calculator

1. Select single number, range, or custom list mode.
2. Enter a positive whole number for n.
3. Use range mode to compare many consecutive values.
4. Use list mode for comma, space, or line separated values.
5. Check the coprime option when n is small.
6. Press the calculate button.
7. Review the factorization, substitution, and step output.
8. Download the results as CSV or PDF when needed.

Understanding Euler Totient Values

Euler’s totient function is a core idea in elementary number theory. It counts how many positive integers from one to n are relatively prime to n. Two numbers are relatively prime when their greatest common divisor is one. This simple count appears in modular arithmetic, cryptography, fraction reduction, cyclic groups, and many contest problems.

Why Prime Factors Matter

The fastest hand method uses distinct prime factors. After n is split into prime powers, each different prime removes a fixed share of numbers. A factor of two removes every second number. A factor of three removes every third number. The product rule adjusts n once for each distinct prime factor. Repeated powers do not add new removal fractions. For example, 72 equals 2^3 × 3^2, yet only primes 2 and 3 appear in the product formula.

What This Calculator Shows

This calculator is designed to teach the process, not only return an answer. It shows factorization, the formula substitution, each prime adjustment, the final totient, and the coprime density. You can process one value, a whole range, or a custom list. For small inputs, you can also show the actual numbers counted by the function. That makes the result easier to verify.

Practical Number Theory Uses

Totient values are useful whenever remainders repeat in cycles. They support Euler’s theorem, which says that a number coprime to n has a power that leaves remainder one modulo n. This is used in modular inverses and public key systems. The function also helps describe how many reduced fractions have a given denominator. In group theory, it counts generators of some cyclic structures.

Reading Results Carefully

A prime number p always has phi value p minus one. The value for one is defined as one. For every n greater than two, the result is even. Large composite numbers can still have high totient values when they have few distinct prime factors. Numbers with many small prime factors usually lose a larger share of candidates. Use the step output to see exactly where each reduction happens.

Checking Work By Hand

For learning, compare answers with a coprime list. Then repeat the formula. Matching values confirm each prime factor adjustment with accuracy and confidence.

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