A cube ending in eleven

MediumProbability Brainteasers~7m

An integer $x$ is chosen uniformly at random from $\{1, 2, \dots, 10^{100}\}$ . What is the probability that $x^3$ ends in the digits $11$ (i.e. $x^3 \equiv 11 \pmod{100}$ )?

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Only $x \bmod 100$ matters (it's uniform). Solve $x^3\equiv 11 \pmod{100}$ via CRT (mod $4$ and mod $25$ ).
Cubing is a bijection mod $25$ (one solution) and forces one class mod $4$ - one residue mod $100$ .

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A cube ending in eleven

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