jmc

Since $xy=144$, $x$ can be any positive divisor of 144. Since $y=\frac{144}{x}$, there is exactly one positive integer $y$ for each positive integer $x$. We can count the ordered pairs by counting the values of $x$, which are the divisors of 144: $$144=2^4\cdot 3^2\Rightarrow t(144)=(4+1)(2+1)=\boxed{15}.$$