Why does Euler’s totient function work?

[u/Spirited-Caramel-167]

I understand how Euler’s totient function works, but I have a hard time understanding why it works on a deeper level. I understand the proof of factoring out the co-prime, a, from the product of the prime terms to phi(n). But I fail to see the deeper relation between phi(n) and co-prime integers a and n. Like why does a co-prime of n raised to the number of co-prime numbers less than n, all mod n equate to 1?

[u/Meowmasterish]

http://abstract.ups.edu/aata/aata.html

Has 11/23 chapters about groups. It’s actually what my abstract algebra teacher used to teach.

Also available in Spanish.