Show that (36m + n)(m + 36n) is not a power of 2 for any positive integers m, n.
Solution
Assume there is a solution. Take m ≤ n and the smallest possible m. Now (36m + n) and (m + 36n) must each be powers of 2. Hence 4 divides n and 4 divides m. So m/2 and n/2 is a smaller solution with m/2 < m. Contradiction.
© John Scholes
jscholes@kalva.demon.co.uk
11 Apr 2002