# How To Calculate 3^233 Mod 353 ? Answer Is 238

## 2 Answers

Ankur Jrm answered

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Oddman answered
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.
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