= Solution
<Repeated squaring> computes $a,a^2,a^4,a^8,\ldots$ by successively squaring. The <binary representation> of the exponent specifies which of these values to multiply. For <modular exponentiation>, reduce after every multiplication; only a number of steps proportional to the <logarithm> of the exponent is needed.
Here $13\equiv2\pmod{11}$ and $17=16+1$. Successive squares give
$$
2^2\equiv4,\quad2^4\equiv16\equiv5,\quad2^8\equiv25\equiv3,\quad2^{16}\equiv9\pmod{11}.
$$
Therefore
$$
\boxed{13^{17}\equiv2^{16}2\equiv9\cdot2\equiv7\pmod{11};\quad\text{remainder }7.}
$$
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