(In reply to
solution by Charlie)
This puzzle can also be solved in terms of methodologies as furnished below.
At the outset, it is observed that 19 is relatively prime to 92. Accordingly, by Euler's Theorem, 19^S(92)=1(Mod 92), where S(x) for any positive integer n denotes the number of integers less than n and prime to n.
So, S(92) = 92(1 - 1/2)(1 - 1/23) = 44, as 2 and 23 constitute the only prime factors of 92.
Hence, 19^44 = 1 (Mod 92)
Or, 19^88 = 1(Mod 92).
Also, 19^4 = (361)^2 (Mod 92) = (-7)^2 (Mod 92)=49(Mod 92).
Consequently, 19^92 =(19^88)*(19^4) = 1* 49(Mod 92)
=49(Mod 92), giving the required remainder as 49.
Alternate Solution
