Find the remainder when 529^643 is divided by 221. ...how to solve this using only remainder theorem?

Hi Kushal,
As we can see here that 221 has factors of 13*17 and 529 has factors of 23*23, Now we will apply here Chinese remainder theorem, according to which, we will calculate separate reminders with 13 and 17 by eular and that will be 9 and 8 respectively. Now we will find the integral solution for the equation 13x+17y = 1. Hence x = 4 and y = -3. Now actual remainder will be 13*4*8 - 17*3*9 = -43 or 178. To learn how to solve questions based on Chinese remainder theorem please visit the link too. link: https://cracku.in/cat/quant-and-di/number-systems/chinese-remainder-theorem-video