If $2^{2013}-2^{2012}-2^{2011}+2^{2010} = k \cdot 2^{2010}$, find $k$ Hey this is just a question i was having fun with but couldnt solve for some reason. Would love if you can help me solve it thankyou!:

If $2^{2013}-2^{2012}-2^{2011}+2^{2010} = k\times2^{2010}$.  find k
Well there is of course the option of simplifying via a calculator but if you want to do it by hand, first I would pull out the factor of $2^{2010}$ out of the left hand side to get
$$2^{2010}(2^3-2^2-2+1)=k\cdot 2^{2010}$$
Then simplify and divide both sides by $2^{2010}$ to get
$$\frac{2^{2010}\cdot 3}{2^{2010}}=k$$
Which yields
$$3=k$$
As originally written, the answer was 4

2^2013 -2^2012 -2^2011 +2^2011 = k * 2^2010

2^2013 -2^2012 (-2^2011 +2^2011) = k * 2^2010

2^2013 -2^2012 +0 = k * 2^2010

(2^2010 * 2^3) - (2^2010 * 2^2) = k * 2^2010

2^2010 (2^3 - 2^2)
------------------------- = k
2^2010

2^3 - 2^2 = k

8 - 4 = k

4 = k



Now, by the same reasoning, the answer becomes

8-4-2+1 = k

3 = k