To divide $10^{30}$ by 97, we need to find the quotient $q$ and the remainder $r$ such that $10^{30} = 97q + r$, where $0 \le r < 97$.

This is equivalent to finding $10^{30} \pmod{97}$. Since 97 is a prime number, we can use Fermat's Little Theorem, which states that if $p$ is a prime number, then for any integer $a$ not divisible by $p$, we have $a^{p-1} \equiv 1 \pmod{p}$.

In this case, $a = 10$ and $p = 97$. Since 97 does not divide 10, we can apply Fermat's Little Theorem:
$10^{97-1} \equiv 1 \pmod{97}$
$10^{96} \equiv 1 \pmod{97}$

Now we want to find $10^{30} \pmod{97}$. We can use the property of modular arithmetic that if $a \equiv b \pmod{m}$, then $a^k \equiv b^k \pmod{m}$ for any positive integer $k$.

We will calculate the powers of 10 modulo 97:
$10^1 \equiv 10 \pmod{97}$
$10^2 \equiv 100 \pmod{97}$
Since $100 = 1 \times 97 + 3$, we have:
$10^2 \equiv 3 \pmod{97}$

Now we can use this to calculate higher powers:
$10^4 = (10^2)^2 \equiv 3^2 \pmod{97}$
$10^4 \equiv 9 \pmod{97}$

$10^8 = (10^4)^2 \equiv 9^2 \pmod{97}$
$10^8 \equiv 81 \pmod{97}$

$10^{16} = (10^8)^2 \equiv 81^2 \pmod{97}$
$81^2 = 6561$
To find $6561 \pmod{97}$, we can divide 6561 by 97:
$6561 \div 97 \approx 67.639$
$97 \times 67 = 6499$
$6561 - 6499 = 62$
So, $10^{16} \equiv 62 \pmod{97}$.

Now we want to find $10^{30}$. We can write 30 in binary: $30 = 16 + 8 + 4 + 2$.
So, $10^{30} = 10^{16} \times 10^8 \times 10^4 \times 10^2$.

$10^{30} \equiv 10^{16} \pmod{97} \times 10^8 \pmod{97} \times 10^4 \pmod{97} \times 10^2 \pmod{97}$
$10^{30} \equiv 62 \times 81 \times 9 \times 3 \pmod{97}$

Let's compute this step by step:
$62 \times 81 \pmod{97}$:
$62 \times 81 = 5022$
$5022 \div 97 \approx 51.77$
$97 \times 51 = 4947$
$5022 - 4947 = 75$
So, $62 \times 81 \equiv 75 \pmod{97}$.

Now multiply by 9:
$75 \times 9 \pmod{97}$:
$75 \times 9 = 675$
$675 \div 97 \approx 6.95$
$97 \times 6 = 582$
$675 - 582 = 93$
So, $75 \times 9 \equiv 93 \pmod{97}$.

Now multiply by 3:
$93 \times 3 \pmod{97}$:
$93 \times 3 = 279$
$279 \div 97 \approx 2.87$
$97 \times 2 = 194$
$279 - 194 = 85$
So, $93 \times 3 \equiv 85 \pmod{97}$.

Therefore, $10^{30} \equiv 85 \pmod{97}$.

This means that when $10^{30}$ is divided by 97, the remainder is 85.

To find the quotient, we have $10^{30} = 97q + 85$.
So, $97q = 10^{30} - 85$.
$q = \frac{10^{30} - 85}{97}$.

The quotient is a very large number. It is generally sufficient to provide the remainder in modular arithmetic problems, unless explicitly asked for the quotient.

The quotient is $\frac{10^{30} - 85}{97}$.

The final answer is $\boxed{quotient = \frac{10^{30} - 85}{97}, remainder = 85}$.
