A number theory problem by Ilham Saiful Fauzi

Use the following divisor-sum formula is used to determine the value of $A$ : $\prod_k \left(\sum\limits_{i=0}^{a_k} p_k^i \right)$ where $p_k$ is the prime in the prime factorization. Observe that $\begin{array}{rl} 819 &= 3^2 \cdot 7 \cdot 13\\ &= 3^2 \cdot 7 \cdot \left(1 + 3 + 3^2\right)\\ &= 63 \cdot \left(1 + 3 + 3^2\right)\\ &= \left(1 + 2 + 2^2 + 2^3 + 2^4 + 2^5\right)\left(1 + 3 + 3^2\right) \end{array}$ Thus, $A = 2^5 \cdot 3^2 = \boxed{288}$ . You can verify this by working backward with that formula.