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Question: Suppose today is Wednesday. What day of the week will it be $10^{{10}^{10}}$ days from now?

Answer: This is what I have tried so far. I know that you need to find $10^{{10}^{10}}\text{mod} 7$ to solve this problem. If $10^{{10}^{10}} \text{mod}7=0$, then it will be Wednesday, if $10^{{10}^{10}}\text{mod}7=1$, it will be Thursday, and so on. I tried breaking up the large exponent as follows: $10^{{10}^{10}}\text{mod}7=(10^{{10}^5} \text{mod}7\cdot 10^{{10}^5}\text{mod7})\text{mod}7$. However, I do not know how to find $10^{{10}^5} \text{mod}7$. The answer to this problem is Sunday, so that means that $10^{{10}^{10}}\text{mod}7=4$.

J. W. Tanner
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Math Fanatic
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    Hint: the sequence $10^i \bmod 7$ has a period; i.e., there's a $j$ such that $10^{i+j}\equiv 10^i\bmod 7$ for all $i$. Then can you see how finding $10^{10}\bmod j$ would help you? – Steven Stadnicki Nov 04 '20 at 23:16
  • How do you define days and weeks if earth has stopped existing an eternity ago? – Servaes Nov 04 '20 at 23:22
  • @ Steven Stadnicki I think that Joshua Wang used your technique in the answer below. However, I don't know where $10^{10}\text{mod}j$ comes from. – Math Fanatic Nov 04 '20 at 23:28

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Note that $10^{6}\equiv 1\pmod{7}$. Then, to solve this problem we would like to find $10^{10}\pmod{6}$. This is equivalent to $(-2)^{10}\equiv 1024\equiv 4\pmod 6$. Thus, $10^{10^{10}}\equiv 10^{6k + 4}\pmod{7}$ for some integer $k$. Thus, $10^{10^{10}}\equiv 10000\equiv 4\pmod{7}$. Can you take it from here?

Joshua Wang
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  • @ Joshua Wang I think you are using the technique described above by Steven Stadnicki. Could you explain where $10^{10}\text{mod}6$ comes from? – Math Fanatic Nov 04 '20 at 23:41
  • Because $10^{0}\equiv 10^{6}\equiv 1\pmod{7}$, the sequence of powers of $10$ $\pmod{7}$ repeats every $6$ increments. Therefore, if we can just find what position in the cycle the exponent $10^{10}$ is at, we can find $10^{10^{10}}\pmod{7}$. – Joshua Wang Nov 04 '20 at 23:43