MadeEasy Test Series 2018: General Aptitude - Modular Arithematic

Question

The value of the expression $13^{88} \text{(mod 19)},$ in the range $0$ to $18,$ is ________.

Comments

Got it thanks.

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sumit goyal 1  No :P

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$13^{88} \mod 19 = 13^{72}.13^{16}\mod 19 = 1.(13^{4})^{4} \mod 19 = 4^4 \mod 19 = 9$
(Using Fermat's theorem: $\color{red}{a^{p-1} \equiv 1 \mod p}$)

Answer 1

13^88/19 = (-6)^88/19  (13 could be written as 19-6 which gives -6 as a remainder)

               = ((-6)^2)^44/19 = 36^44/19

               = (38-2)^44/19

              = (-2)^44/19 = 16^11/19 = (-3)^11/19

              = -3 * (-3)^10/19 = -3 * 9^5/19 = -3 * 9 * 9^4/19 = -3 * 9 * (81)^2/19

              = -3 * 9 * (76-5)^2/19 = -3 * 9 * (-5)^2/19 = -3 * 9 * 25 /19

             =  -27 * 6/19 = -8 * 6/19 = -48 /19 = -10/19 = (-19 + 9 ) / 19 = 9 is the remainder