Question 169
a) Which of the numbers is greater: $2^{100}+3^{100}$ or $4^{100}$?
b) Let $x$ and $y$ be natural numbers such that
$$
2^{x} \cdot 3^{y}=\left(24^{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{60}}\right) \cdot\left(24^{\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{60}}\right)^{2} \cdot\left(24^{\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\cdots+\frac{1}{60}}\right)^{3} \cdot \ldots \cdot\left(24^{\frac{1}{60}}\right)^{59}
$$
Determine the value of $x+y$.
b) Let $x$ and $y$ be natural numbers such that
$$
2^{x} \cdot 3^{y}=\left(24^{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{60}}\right) \cdot\left(24^{\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{60}}\right)^{2} \cdot\left(24^{\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\cdots+\frac{1}{60}}\right)^{3} \cdot \ldots \cdot\left(24^{\frac{1}{60}}\right)^{59}
$$
Determine the value of $x+y$.