[Number Theory]
Problem: Let $T=$ 21. The number $20^{T} \cdot 23^{T}$ has $K$ positive divisors. Compute the greatest prime factor of $K$.

Solution: Write $20^{T} \cdot 23^{T}$ as $2^{2 T} \cdot 5^{T} \cdot 23^{T}$. This number has $K=(2 T+1)(T+1)^{2}$ positive divisors. With $T=21, K=43 \cdot 22^{2}$. The greatest prime factor of $K$ is $\mathbf{4 3}$.

Final answer: 43