Heres the problem: Use Euler's Theorem or anything else needed to find the least positive x such that 2^(6073) is congruent to x(mod1023)
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Not quite sure what you mean by Euler's Theorem is, but $\displaystyle 2^{6073} = 2^{6070} \cdot 2^3 = (2^{10})^{607} \cdot 2^3 = 1024^{607} \cdot 2^3 \equiv 1^{607} \cdot 2^3 \equiv 8 \mod{1023} $.
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