NUMBER THEORY / COPRIME RESIDUE COUNT
Count positive integers through n that are coprime to n and expose the prime-factor product.
METHOD / WORKED EXAMPLE
Euler's totient counts the reduced residue classes modulo n. The product uses each distinct prime once, so the factor list explains exactly how the count was reduced.
WORKED DEFAULT
For n = 36, the distinct prime factors are 2 and 3. Starting at 36 gives 36 x (1 - 1/2) x (1 - 1/3) = 12, so phi(36) = 12.
READ THE RESULT
The totient is a count, not the list of coprime values. Repeated powers of an existing prime affect n but do not add another distinct product factor.
ASSUMPTIONS AND LIMITS
This deterministic browser calculation is educational number theory; it is not a cryptographic key generator or security review.
COMMON QUESTIONS
The standard count here uses positive integers from 1 through n. Zero has gcd(0,n) = n, so it is coprime only in the exceptional modulus-one sense and is not added as a separate residue in this count. For n greater than one, the reduced residue system is commonly represented within 1 through n-1.
Euler's product runs over distinct primes dividing n. Removing multiples of p excludes the share divisible by that prime, while higher powers do not define a new divisibility category. Their effect is already present through the original size of n. For example, 2 and 3 each appear once in the product for 36 despite both being squared.
It can support number-theory study, but secure cryptography requires carefully generated large primes, protected randomness, vetted libraries, side-channel resistance, parameter standards, and key-management controls. This browser tool is bounded at one billion and exposes factors for transparency. It must not generate or validate production cryptographic keys, moduli, or security guarantees.
RELATED TOOLS
Use boundary
Factor n into distinct primes, then apply the multiplicative product by subtracting one p-th of the running count for each distinct prime divisor.
phi(n) = n product over distinct p dividing n of (1 - 1/p).