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Euler's totient calculator φ(n)

Compute Euler's totient φ(n) — the count of integers from 1 to n that are coprime to n (share no factor but 1). It factorises n into primes and applies φ(n) = n·∏(1−1/p) over its distinct prime factors, showing the factorization, the formula and the number of coprimes. φ underlies RSA, Euler's theorem and the order of the multiplicative group mod n.

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The Euler's totient calculator φ(n) turns n (positive integer) into φ(n), Count of coprimes in 1…n, Prime factorization of n, instantly and for free. For instance, with n (positive integer) = 36 it returns φ(n) = 12, Count of coprimes in 1…n = 12 and Prime factorization of n = 2² × 3².

How to use it

  1. Enter your values: n (positive integer).
  2. Read the result instantly: φ(n), Count of coprimes in 1…n, Prime factorization of n.

Frequently asked questions

How does the Euler's totient calculator φ(n) work?

It takes n (positive integer) and derives φ(n), Count of coprimes in 1…n and Prime factorization of n from them. The calculation is live as you type, so the result updates on every change.

Which values does the calculator ask for?

A single value: n (positive integer). Nothing else is required — no account, no file upload.

What does a typical calculation look like?

With n (positive integer) = 36, the calculator returns φ(n) = 12, Count of coprimes in 1…n = 12 and Prime factorization of n = 2² × 3². Those figures come from running this exact tool, so you can reproduce them by entering the same values.

How much does the result change with different inputs?

It moves a lot. Using n (positive integer) = 72 instead, φ(n) goes from 12 to 24 — which is why it is worth testing a few scenarios rather than trusting a single figure.

What does it give for smaller values?

Scaled down to n (positive integer) = 18, φ(n) comes out at 6. The relationship is worth checking at both ends before you rely on a single result.

When would I actually use this?

Reading a number written another way: a Roman numeral on a monument, a hexadecimal colour, a binary byte, or a fraction that needs reducing.

What is the most common mistake?

Assuming a decimal fraction survives a change of base. A tenth is exact in decimal and infinitely repeating in binary, which is why 0.1 + 0.2 is not 0.3 in most programming languages.

What is the difference between the Euler's totient calculator φ(n) and the e Digits Calculator (Euler's Number)?

This one returns φ(n) and Count of coprimes in 1…n; the e Digits Calculator (Euler's Number) returns Result. That is the whole difference — open the one whose figure you need.

Is there a tool for the next step?

Euler characteristic calculator (V − E + F) is the closest one after this: For a polyhedron or a triangulated surface, χ = V − E + F. A convex polyhedron gives χ = 2; the genus g = (2 − χ)/2 tells you how many handles the surface has (0 = sphere, 1 = torus).

What else is worth having open alongside it?

Prime factorization calculator and Euler's Method Calculator — they come up in the same task often enough to be worth a second tab.

Further reading

All guides →
ExplainerSignificant Figures Explained: Counting, Rounding, and Why They MatterLearn what counts as a significant figure, the rules for leading, trailing, and captive zeros, and how sig figs keep your calculated answers honest.How-toHow to Find Prime Factors: Factor Trees and the Division MethodBreak any whole number into its prime building blocks with a factor tree or repeated division — and write the result as a clean prime factorization with exponents.How-toHow to Multiply and Divide Fractions: Multiply Across, Flip to DivideMultiplying fractions needs no common denominator — just multiply across. Dividing means multiplying by the reciprocal. Learn both with worked examples and how to simplify.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.How-toHow to Round Numbers: Round Half Up, Round Half to Even, and MoreRound to any decimal place or number of significant figures, and learn when to use round half up versus round half to even to avoid bias.How-toHow to Add and Subtract Fractions: Common Denominator, Combine, SimplifyYou can only add or subtract fractions that share a denominator. Learn to find a common denominator, combine the numerators, and simplify the result, with a worked example.