The largest term in the binomial expansion of $(1 + \frac{1}{2})^{31}$ is of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is the value of $b$? 

As an example of a binomial expansion, the binomial expansion of an expression of the form $(x + y)^3$ is the sum of four terms: 
\[ x^3 + 3x^2y + 3xy^2 + y^3. \]

Let's think step by step and output the final answer within \boxed{}.
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