To find the number of positive divisors, we first need to find the prime factorization of the number 68600.
We can write 68600 as $686 \times 100$.
First, let's factorize 100:
$100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2$.
Next, let's factorize 686:
$686 = 2 \times 343$.
Recognize that 343 is a perfect cube: $343 = 7^3$.
So, $686 = 2 \times 7^3$.
Now, combine the prime factors:
$68600 = (2 \times 7^3) \times (2^2 \times 5^2)$.
Group the powers of the same prime bases:
$68600 = 2^{1+2} \times 5^2 \times 7^3 = 2^3 \times 5^2 \times 7^3$.
The formula for the number of divisors of a number $N = p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}$ is $(a_1+1)(a_2+1)\dots(a_k+1)$.
For $68600 = 2^3 \times 5^2 \times 7^3$, the exponents are 3, 2, and 3.
Number of divisors = $(3+1) \times (2+1) \times (3+1)$.
Number of divisors = $4 \times 3 \times 4$.
Number of divisors = 48.