Question

Find the remainder when 3^{60} is divided by 242 using remainder theory

Answer 1

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243

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For any power of 3 unit digits would be 1, 3,7,or 9. Also if you notice, after every 4th power of 3, the unit digit would repeat itself. 
Therefore, in the question 3^243 (or 3^(240+3)) would have unit digit of 7
Hence when divided by 5, it will give remainder =2.