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?
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.
.
.
.
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!
