Euler's Totient Function (φ) Calculator
Enter an integer N (1 to 1,000,000) to instantly find the number of integers from 1 to N that are coprime to N (Euler's totient function φ(N)), with prime factorization and the formula used. Includes a φ(n) reference table for 1-100.
φ(n) reference table for 1-100
A reference table showing the prime factorization and Euler totient value for every integer from 1 to 100.
| N | Factorization | φ(n) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 2 | 1 |
| 3 | 3 | 2 |
| 4 | 2² | 2 |
| 5 | 5 | 4 |
| 6 | 2 × 3 | 2 |
| 7 | 7 | 6 |
| 8 | 2³ | 4 |
| 9 | 3² | 6 |
| 10 | 2 × 5 | 4 |
| 11 | 11 | 10 |
| 12 | 2² × 3 | 4 |
| 13 | 13 | 12 |
| 14 | 2 × 7 | 6 |
| 15 | 3 × 5 | 8 |
| 16 | 2⁴ | 8 |
| 17 | 17 | 16 |
| 18 | 2 × 3² | 6 |
| 19 | 19 | 18 |
| 20 | 2² × 5 | 8 |
| 21 | 3 × 7 | 12 |
| 22 | 2 × 11 | 10 |
| 23 | 23 | 22 |
| 24 | 2³ × 3 | 8 |
| 25 | 5² | 20 |
| 26 | 2 × 13 | 12 |
| 27 | 3³ | 18 |
| 28 | 2² × 7 | 12 |
| 29 | 29 | 28 |
| 30 | 2 × 3 × 5 | 8 |
| 31 | 31 | 30 |
| 32 | 2⁵ | 16 |
| 33 | 3 × 11 | 20 |
| 34 | 2 × 17 | 16 |
| 35 | 5 × 7 | 24 |
| 36 | 2² × 3² | 12 |
| 37 | 37 | 36 |
| 38 | 2 × 19 | 18 |
| 39 | 3 × 13 | 24 |
| 40 | 2³ × 5 | 16 |
| 41 | 41 | 40 |
| 42 | 2 × 3 × 7 | 12 |
| 43 | 43 | 42 |
| 44 | 2² × 11 | 20 |
| 45 | 3² × 5 | 24 |
| 46 | 2 × 23 | 22 |
| 47 | 47 | 46 |
| 48 | 2⁴ × 3 | 16 |
| 49 | 7² | 42 |
| 50 | 2 × 5² | 20 |
| 51 | 3 × 17 | 32 |
| 52 | 2² × 13 | 24 |
| 53 | 53 | 52 |
| 54 | 2 × 3³ | 18 |
| 55 | 5 × 11 | 40 |
| 56 | 2³ × 7 | 24 |
| 57 | 3 × 19 | 36 |
| 58 | 2 × 29 | 28 |
| 59 | 59 | 58 |
| 60 | 2² × 3 × 5 | 16 |
| 61 | 61 | 60 |
| 62 | 2 × 31 | 30 |
| 63 | 3² × 7 | 36 |
| 64 | 2⁶ | 32 |
| 65 | 5 × 13 | 48 |
| 66 | 2 × 3 × 11 | 20 |
| 67 | 67 | 66 |
| 68 | 2² × 17 | 32 |
| 69 | 3 × 23 | 44 |
| 70 | 2 × 5 × 7 | 24 |
| 71 | 71 | 70 |
| 72 | 2³ × 3² | 24 |
| 73 | 73 | 72 |
| 74 | 2 × 37 | 36 |
| 75 | 3 × 5² | 40 |
| 76 | 2² × 19 | 36 |
| 77 | 7 × 11 | 60 |
| 78 | 2 × 3 × 13 | 24 |
| 79 | 79 | 78 |
| 80 | 2⁴ × 5 | 32 |
| 81 | 3⁴ | 54 |
| 82 | 2 × 41 | 40 |
| 83 | 83 | 82 |
| 84 | 2² × 3 × 7 | 24 |
| 85 | 5 × 17 | 64 |
| 86 | 2 × 43 | 42 |
| 87 | 3 × 29 | 56 |
| 88 | 2³ × 11 | 40 |
| 89 | 89 | 88 |
| 90 | 2 × 3² × 5 | 24 |
| 91 | 7 × 13 | 72 |
| 92 | 2² × 23 | 44 |
| 93 | 3 × 31 | 60 |
| 94 | 2 × 47 | 46 |
| 95 | 5 × 19 | 72 |
| 96 | 2⁵ × 3 | 32 |
| 97 | 97 | 96 |
| 98 | 2 × 7² | 42 |
| 99 | 3² × 11 | 60 |
| 100 | 2² × 5² | 40 |
Tips
- Euler's totient function φ(n) counts the integers from 1 to n that are coprime to n. For example, φ(9) = 6 (1, 2, 4, 5, 7, and 8 are coprime to 9).
- When n is a prime p, φ(p) = p − 1, because every number from 1 to p except p itself is coprime to a prime.
- When n is the product of two distinct primes p and q, φ(pq) = (p−1)(q−1) — a property used directly in RSA key generation.
- The formula φ(n) = n × Π(1 − 1/p) is a product over the distinct prime factors p of n; the exponent of each prime does not affect the result.
- This tool supports the range 1 to 1,000,000 and computes φ(n) instantly using trial-division factorization.
Frequently Asked Questions
Side Note — The 18th-century function that counts what's coprime
Euler's totient function was introduced around 1763 by the Swiss mathematician Leonhard Euler, originally while generalizing Fermat's little theorem to composite numbers. The now-standard symbol φ (phi) was adopted later by other mathematicians who systematized the notation.
Counting coprime numbers may sound like a simple exercise, but φ(n) has several elegant properties. For instance, summing φ(d) over every divisor d of n always equals n itself (Σφ(d) = n) — one of the fundamental identities in elementary number theory.
Today, φ(n) plays an essential role in RSA cryptography. RSA uses the product of two large primes n = pq as part of the public key, and computing the private key requires φ(n) = (p−1)(q−1). The fact that φ(n) cannot be computed without factoring n (at least with currently known algorithms) is one of the pillars underlying RSA's security.