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Let n be a positive integer and let k be the number of positve integers less than 2^n.If 2^n is congruent to 3 (mod 13), then what is the remainder when k is divided by 13?

 Oct 19, 2020
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Let us see if I understand your question:

 

Let "n" be a positive integer. I will choose "n" to be = 4. So, 2^n = 2^4 =16, and 16 mod 13 =3. And let "k" be the number of positive integers LESS than 2^4 or 16. In other words, "k" will range from 1 to 15. So, the question is: what is the remainder when"k" is divided by 13? Well, if "k" is from 1 to 15, then you have the following:

 

1 mod 13 = 1 - the remainder

2 mod 13 = 2 - the remainder

3 mod 13 = 3 - the remainder.

4 mod 13 = 4 - the remainder.

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13 mod 13 = 0

14 mod 13 = 1 - the remainder

15 mod 13 = 2 -the remainder. And so on. And this pattern will continue for any "n" you choose.

 

And that is the way I read it! 

 Oct 19, 2020

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