AoPS Wiki talk:Problem of the Day/July 25, 2011

Find the remainder when $2^{2011}$ is divided by 11.

Solution 1

Let's find a pattern for remainders when this number is divided by 11.

$2^1$ has a remainder of 2.
$2^2$ has a remainder of 4.
$2^3$ has a remainder of 8.
$2^4$ has a remainder of 5.
$2^5$ has a remainder of 10.
$2^6$ has a remainder of 9.
$2^7$ has a remainder of 7.
$2^8$ has a remainder of 3.
$2^9$ has a remainder of 6.
Finally, $2^{10}$ has a remainder of 1.

Starting from now on, this pattern will keep repeating. Every positive integer power of $2^{10}$ will have a remainder of 1 once divided by 11. This includes $2^{2010}$ . So, $2^{2011}$ has a remainder of $\boxed{2}$ when divided by 11.

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AoPS Wiki talk:Problem of the Day/July 25, 2011

Solution 1