To find the quotient and remainder of $10^{30}$ divided by $97$, we can use **Fermat's Little Theorem**.

### 1. Identify the Modulo
Fermat's Little Theorem states that if $p$ is a prime number, then for any integer $a$ not divisible by $p$:
$$a^{p-1} \equiv 1 \pmod{p}$$
Here, $p = 97$, which is a prime number. Therefore:
$$10^{96} \equiv 1 \pmod{97}$$

### 2. Calculate $10^{30} \pmod{97}$
While Fermat's Little Theorem doesn't directly solve for the 30th power, we can use the method of successive squaring to find $10^{30} \pmod{97}$:

*   $10^1 \equiv 10 \pmod{97}$
*   $10^2 = 100 \equiv 3 \pmod{97}$
*   $10^4 = (10^2)^2 = 3^2 = 9 \pmod{97}$
*   $10^8 = (10^4)^2 = 9^2 = 81 \equiv -16 \pmod{97}$
*   $10^{16} = (10^8)^2 = (-16)^2 = 256$
    *   $256 \div 97 \approx 2.63 \dots$
    *   $97 \times 2 = 194$
    *   $256 - 194 = 62 \equiv -35 \pmod{97}$

Now, combine these to form $10^{30} = 10^{16} \cdot 10^8 \cdot 10^4 \cdot 10^2$:
*   $10^{30} \equiv (-35) \cdot (-16) \cdot 9 \cdot 3 \pmod{97}$
*   $(-35) \cdot (-16) = 560$
    *   $560 \div 97 \approx 5.77$
    *   $97 \times 5 = 485$
    *   $560 - 485 = 75 \equiv -22 \pmod{97}$
*   $(-22) \cdot (9 \cdot 3) = -22 \cdot 27$
*   $-22 \cdot 27 = -594$
    *   $-594 \div 97 \approx -6.12$
    *   To find the positive remainder: $-594 + (97 \times 7) = -594 + 679 = 85$

So, the remainder is **85**.

### 3. Calculate the Quotient
The quotient $Q$ is given by $\lfloor \frac{10^{30}}{97} \rfloor$.
Since $10^{30} = 97 \times Q + 85$:
$$Q = \frac{10^{30} - 85}{97}$$

Calculating this value:
$10^{30} = 1,000,000,000,000,000,000,000,000,000,000$
$10^{30} - 85 = 999,999,999,999,999,999,999,999,999,915$

Dividing by 97:
**Quotient:** $10,309,278,350,515,463,917,525,773,195$
**Remainder:** $85$
