TITLE: Prove that the expression cannot be a power of 2
QUESTION [0 upvotes]: I have been boggled by this question for a while as well.
Prove that
$$(2a+b)(2b+a)=2^c$$
Is impossible.
I know that if a and b do exist then they must be even. I am trying to use this fact to contradict the statement. I haven also tried rewriting a and b as products of powers of twos and a odd factor

REPLY [0 votes]: Presumably $a,b,c$ are positive integers. Without loss of generality, suppose $a\le b$. The given condition implies that both $2a+b$ and $a+2b$ are powers of $2$. Hence $a+2b=2^k(2a+b)$ for some nonnegative integer $k$. Yet this is impossible because $a+2b>2^0(2a+b)$ and $a+2b<2^k(2a+b)$ for every $k\ge1$. Hence the given condition cannot be satisfied.
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