2021 GMC 10B Problems/Problem 6

Problem

How many possible ordered pairs of nonnegative integers $(a,b)$ are there such that $2a+3^b=4^{ab}$ ?

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3\qquad\textbf{(E)} ~4$

Solution

Note that the LHS is always odd while the RHS is always even for $a, b > 0$ . Since this is obviously never true, the only possible solution case is $a, b = 0$ . Since $3^0 = 4^0 = 1$ , $(0, 0)$ is the only solution, thus there is only $\textbf{(B)}~1$ solution.

~pineconee

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2021 GMC 10B Problems/Problem 6

Problem

Solution