Totatives list calculator

This calculator produces the complete ordered list rather than only reporting Euler's totient count. Enter a positive integer to receive the original value, the number of matching integers, and the totatives themselves. It is useful for checking modular arithmetic exercises, exploring reduced residue systems, and seeing exactly which values contribute to the Euler totient function.

What the totatives list represents

The totatives of n are precisely the integers from 1 through n whose greatest common divisor with n equals 1. Saying that two numbers have greatest common divisor 1 is the same as saying they are coprime or relatively prime. For example, a candidate is excluded if it shares any prime factor with n, even when the candidate does not divide n. The returned array is sorted in increasing order because candidates are tested from 1 upward. The value 1 always appears, since it is coprime to every positive integer. The endpoint n normally does not appear because gcd(n, n) equals n; the special case is n = 1, for which the list is [1]. The accompanying count equals the length of the list and therefore equals Euler's totient function phi(n). This capability exposes the actual members, while a totient-only calculator is preferable when you need only the count for a very large integer. The distinction matters in modular arithmetic, where the members themselves form the reduced residue system modulo n.

How the calculation works

The calculator validates n before doing any arithmetic. It accepts a whole number or an integer written as a plain decimal string, rejects fractions and nonnumeric values, and reports an input error when n is below 1. It also enforces the published upper bound so that producing a potentially large JSON array remains predictable in both the browser and the API. After validation, the algorithm considers every integer from 1 to n. For each candidate it applies the Euclidean algorithm: repeatedly replace the larger pair with the divisor and remainder until the remainder becomes zero. The last nonzero divisor is the greatest common divisor. A candidate enters the result only when that divisor is 1. This procedure is exact integer arithmetic; it uses no approximation, factor database, network request, random choice, or clock. Consequently the same input always yields the same ordered output. The count is derived from the completed array rather than calculated separately, preventing disagreement between the displayed list and its reported size.

Using the result correctly

Use the list when your next step depends on the individual residue classes rather than merely their quantity. In elementary number theory, it provides a direct way to verify which numbers are invertible modulo n: every listed value has a multiplicative inverse modulo n, and every omitted value does not. In cryptography lessons, the output can help demonstrate why a multiplier must be coprime to a modulus, though it is an educational arithmetic result rather than a key-generation system. You can also compare the returned count with a hand calculation of Euler's totient formula to check factorization work. Remember that coprimality is a relationship, not a claim that each listed number is prime. Composite values may appear whenever they share no prime factor with n. For instance, a composite candidate can be a totative of a prime modulus. For automated use, read the totatives array directly and treat count as a convenient summary. If you only need to know whether one particular pair is coprime, a pairwise coprime checker avoids constructing the full list.