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kaliprasad
Well-known member
- Mar 31, 2013
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Prove that the last 6 digits of 7^10000 is 000001
Prove that the last 6 digits of 7^10000 is 000001
neater than my solution. I shall post mine one week later so that others can post\[\begin{array}{}
7^4 &=& 2401 &\equiv& 1 \pmod{400} \\
7^{100} &=& (7^4)^{25} &=& (400k+1)^{25} &=& ...\ +\ 25 \cdot 400k + 1 &\equiv& 1 \pmod{10000} \\
7^{10000} &=& (7^{100})^{100} &=& (10000m + 1)^{100} &=& ...\ +\ 100\cdot 10000m + 1 &\equiv& 1 \pmod{1000000} \\
\blacksquare
\end{array}
\]
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I have done the needful.Hello Kali,
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For the convenience of our members, we prefer that they not have to follow links, but that the solution(s) be posted here. This makes it easier on the majority.![]()