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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 =

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

for the following, we are going to temporarily drop the factor

Mod[331^4, 353] =

The answer to your question is

Mod[3^233, 353] = Mod[217*265*176*243, 353] = 248

Yes, the answer is

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.