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2016-12-21 12:48:22

Ans: $2$

$13^{73}\equiv (11+2)^{73}\equiv 2^{73} \\ \equiv \left(2^5\right)^{14}×2^3 \equiv (32)^{14}×2^3 \\

\equiv (-1)^{14}×8 \equiv 8 \pmod{11} ~~\cdots(a)$

$14^{3}\equiv (11+3)^{3}\equiv 3^{3} \\ \equiv 5 \pmod{11} ~~\cdots(b)$

From $(a)$ and $(b)$

$13^{73}+14^3 \equiv (8+5) \equiv 13 \equiv 2 \pmod{11}$

Note: click here to understand about congruence relation.

$13^{73}\equiv (11+2)^{73}\equiv 2^{73} \\ \equiv \left(2^5\right)^{14}×2^3 \equiv (32)^{14}×2^3 \\

\equiv (-1)^{14}×8 \equiv 8 \pmod{11} ~~\cdots(a)$

$14^{3}\equiv (11+3)^{3}\equiv 3^{3} \\ \equiv 5 \pmod{11} ~~\cdots(b)$

From $(a)$ and $(b)$

$13^{73}+14^3 \equiv (8+5) \equiv 13 \equiv 2 \pmod{11}$

Note: click here to understand about congruence relation.

2016-12-19 19:18:52