2021/001)Show that for any positive integers a and b $(36a+b)(a+36b)$ cannot be a power of 2

 Without loss of generality let us assume $a\ge b$. so $(36a+b) \ge (a+36b)$

Let $(36a+b)(a+36b)$ be a power of 2
Then we must have both $36a+b $ and $a + 36b$ powers of 2

So let $36a+b = 2^m $ and $a+36b = 2^n$ where $m \ge n$
So $36a+b + 3a + 36 b = 37(a+b) = 2^n(2^{m-n} + 1)$
So $37$ has to be $2^n$ or $2^{(m-n)} + 1$ and neither is possible as neither 37 not 36 is a power of 2
Hence there is contradiction
Hence it is not possible