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Difference between revisions of "AoPS Wiki talk:Problem of the Day/July 25, 2011"

(Created page with "Find the remainder when <math>2^{2011}</math> is divided by 11.")
 
 
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Find the remainder when <math>2^{2011}</math> is divided by 11.
 
Find the remainder when <math>2^{2011}</math> is divided by 11.
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==Solution 1==
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Let's find a pattern for remainders when this number is divided by 11.
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*<math>2^1</math> has a remainder of 2.
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*<math>2^2</math> has a remainder of 4.
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*<math>2^3</math> has a remainder of 8.
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*<math>2^4</math> has a remainder of 5.
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*<math>2^5</math> has a remainder of 10.
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*<math>2^6</math> has a remainder of 9.
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*<math>2^7</math> has a remainder of 7.
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*<math>2^8</math> has a remainder of 3.
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*<math>2^9</math> has a remainder of 6.
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*Finally, <math>2^{10}</math> has a remainder of 1.
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Starting from now on, this pattern will keep repeating.
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Every positive integer power of <math>2^{10}</math> will have a remainder of 1 once divided by 11.
 +
This includes <math>2^{2010}</math>.
 +
So, <math>2^{2011}</math> has a remainder of <math>\boxed{2}</math> when divided by 11.

Latest revision as of 19:55, 25 July 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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