Question 175
a) For which natural numbers $n$ do there exist natural numbers $x$ and $y$ such that
$$
\operatorname{lcm}(x, y) = n!; \quad \gcd(x, y) = 2009
$$
b) Determine the number of pairs $(x, y)$ for which
$$
\operatorname{lcm}(x, y) = 41!; \quad \gcd(x, y) = 2009; \quad x \le y.
$$
$$
\operatorname{lcm}(x, y) = n!; \quad \gcd(x, y) = 2009
$$
b) Determine the number of pairs $(x, y)$ for which
$$
\operatorname{lcm}(x, y) = 41!; \quad \gcd(x, y) = 2009; \quad x \le y.
$$