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There may be easier ways, but I did it this way. Mod[3^6, 353] = 23. 233 = 38*6 + 5 so Mod[3^233, 353] = Mod[(23^38)*(3^5), 353] for the following, we are going to temporarily drop the factor 3^5 = 243
38 = 2*19, so Mod[23^38, 353] = Mod[Mod[23^2, 353]^19, 353] = Mod[176^19, 353] 19 = 2*9 + 1, so Mod[176^19, 353] = Mod[Mod[176^2, 353]^9*176, 353] = Mod[(265^9)*176, 353] for the following, we are going to temporarily drop the factor 176
9 = 2*4 + 1, so Mod[(265^9), 353] = Mod[(Mod[265^2, 353]^4)*265, 353] = Mod[(331^4)*265, 353] for the following, we are going to temporarily drop the factor 265
Mod[331^4, 353] is within the range of my calculator. Mod[331^4, 353] = 217. The answer to your question is Mod[3^233, 353] = Mod[217*265*176*243, 353] = 248 Yes, the answer is 248, confirmed independently.
