jmc

We can simplify $\frac{19}{t}+\frac{5}{t}$ as $\frac{19+5}{t}$, or $\frac{24}{t}$. So, in order for this expression to have an integral value, 24 must be divisible by $t$. In other words, $t$ must be a factor of 24. Therefore, to find the number of positive integers $t$ that will make the expression have an integral value, we just need to find the number of factors of 24. We know that if $n={p_1}^{e_1}\cdot {p_2}^{e_2}\cdot {p_3}^{e_3}\cdots {p_k}^{e_k}$, where $p_1,p_2...p_k$ are prime numbers, then the number of factors of $n$ is equal to $(e_1+1)(e_2+1)(e_3+1)\cdots (e_k+1)$. We have that the prime factorization of 24 is $2^3\cdot 3^1$, so using the formula above, 24 has $(3+1)(1+1)=\boxed{8}$ factors.