If 2^2011 is divided by 13 what is the remainder???
Oh, 2^2011 and division, we have a true math party here! Now, if I were a number, I'd want to divide by 13. Why? Because it sounds like the unluckiest number to divide by. So, let me see if I can crack this one for you while juggling 13 rubber chickens. *juggling* Ah, after a bit of math magic and clown logic, the remainder is... drumroll, please... 6!
To find the remainder when 2^2011 is divided by 13, we can use the concept of modular arithmetic.
First, let's find the remainder when the powers of 2 are divided by 13:
2^0 mod 13 = 1
2^1 mod 13 = 2
2^2 mod 13 = 4
2^3 mod 13 = 8
2^4 mod 13 = 3
2^5 mod 13 = 6
2^6 mod 13 = 12
2^7 mod 13 = 11
2^8 mod 13 = 9
2^9 mod 13 = 5
2^10 mod 13 = 10
2^11 mod 13 = 7
2^12 mod 13 = 1
We can see that the remainder follows a repeating pattern every 12 powers of 2. So, in order to find the remainder of 2^2011, we can find the remainder of 2011 divided by 12, and then check the corresponding power of 2.
2011 divided by 12 gives a quotient of 167 and a remainder of 7.
Therefore, the remainder when 2^2011 is divided by 13 is the same as the remainder when 2^7 is divided by 13.
2^7 mod 13 = 11
So, the remainder is 11.
To find the remainder when 2^2011 is divided by 13, we can use the concept of modular arithmetic.
First, let's observe the pattern of remainders when we divide powers of 2 by 13. We start with 2^0 = 1 (remainder 1), then 2^1 = 2 (remainder 2), 2^2 = 4 (remainder 4), 2^3 = 8 (remainder 8), 2^4 = 16 (remainder 3), 2^5 = 32 (remainder 6), and so on.
We notice that a cycle starts repeating from 2^0 to 2^3, with remainders 1, 2, 4, 8. This cycle has a length of 4. In other words, if we divide any power of 2 by 13, the remainder will be the same as dividing the power of 2 by 4.
Now, we can simplify the problem by finding the remainder when 2011 is divided by 4. In this case, 2011 divided by 4 gives a quotient of 502 with a remainder of 3.
Since the cycle of remainders repeats every 4 powers, we know that 2^3 has the same remainder as 2^2011. From earlier, we found that 2^3 has a remainder of 8 when divided by 13.
Therefore, the remainder when 2^2011 is divided by 13 is 8.
look at the remainders when dividing powers of 2 by 13:
n 2^n mod13
0 1 1
1 2 2
2 4 4
3 8 8
4 16 3
5 32 6
6 64 12
7 128 11
8 256 9
9 512 5
10 1024 10
11 2048 7
12 4096 1
...
so, you see that 2^n mod 13 = 2^(n+12) mod 13
2011 = 167*12 + 7
so, 2^2011 mod 13 = 2^7 mod 13 = 11