Divisors.

The prime factorization of $1097600$ is $2^{7}*5^{2}*7^{3}.$ Thus there are a total of $(7 + 1)(2 + 1)(3 + 1) = 96$ divisors.

For a divisor to be divisible by $10$ it must have at one factor of $2$ and at least one of $5$ and hence we are looking for the number of divisors of

$\frac{1097600}{10} = 1097600 = 2^{6}*5^{1}*7^{3}.$

There are $(6 + 1)(1 + 1)(3 + 1) = 56$ of these, and thus the probability that a randomly chosen divisor of $1097600$ is divisible by $10$ is

$\dfrac{56}{96} = \dfrac{7}{12}.$

Therefore $a + b = 7 + 12 = \boxed{19}$ .

$1097600=2^{7}.5^{2}.7^{3}$ Its divisors can be expressed as $2^{x}5^{y}7^{z}$

Where $x$ can vary from $0$ to $7$ , $y$ from $0$ to $2$ and $z$ from $0$ to $3$ (All inclusive)

Thus by the Fundamental Principle of Counting, the total number of divisors of the given number = $96$

And any of its divisors divisible by $10$ can be expressed as $2^{x}.5^{y}.7^{z}$

Where $x$ can vary from $1$ to $7$ , $y$ from $1$ to $2$ and $z$ from $0$ to $3$ (All inclusive)

Thus by the Fundamental Principle of Counting, the number of divisors of the given number which are divisible by $10$ = $56$

Hence the required probability = $\frac{56}{96}= \frac{7}{12}$

Thus, $a=7, b=12,$ and $a+b=\boxed{19}$