To find the quotient and remainder of $10^{30}$ divided by $97$, we use modular arithmetic for the remainder and long division techniques for the quotient.

### 1. Finding the Remainder ($R$)
We need to find $10^{30} \pmod{97}$. Since 97 is a prime number, we can use **Fermat's Little Theorem**, which states that $a^{p-1} \equiv 1 \pmod p$.

Thus, $10^{96} \equiv 1 \pmod{97}$. This doesn't help directly for $10^{30}$, so we look for powers of 10 close to 97:
*   $10^2 = 100 \equiv 3 \pmod{97}$
*   $10^{30} = (10^2)^{15} \equiv 3^{15} \pmod{97}$

Now we calculate $3^{15} \pmod{97}$:
*   $3^4 = 81 \equiv -16 \pmod{97}$
*   $3^5 = 243$. Since $97 \times 2 = 194$, $243 - 194 = 49$. So $3^5 \equiv 49 \pmod{97}$
*   $3^{10} = (49)^2 = 2401$.
    *   $2401 \div 97$: $97 \times 20 = 1940$; $2401 - 1940 = 461$. 
    *   $97 \times 4 = 388$; $461 - 388 = 73$.
    *   So $3^{10} \equiv 73 \equiv -24 \pmod{97}$
*   $3^{15} = 3^{10} \times 3^5 \equiv (-24) \times 49 \pmod{97}$
    *   $-24 \times 49 = -1176$
    *   $-1176 \div 97$: $97 \times 12 = 1164$.
    *   $-1176 \equiv -12 \pmod{97}$
    *   $-12 + 97 = 85$.

**Remainder ($R$) = 85**

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### 2. Finding the Quotient ($Q$)
The quotient is the integer part of $\frac{10^{30}}{97}$. We can represent this as a repeating decimal. The fraction $\frac{1}{97}$ has a period of 96 digits.
$\frac{1}{97} \approx 0.01030927835051546391752577319587...$

To get the quotient for $\frac{10^{30}}{97}$, we shift the decimal point 30 places to the right and take the integer portion.

**The Quotient ($Q$) is:**
**10,309,278,350,515,463,917,525,773,195**

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### Summary
*   **Quotient:** $10,309,278,350,515,463,917,525,773,195$
*   **Remainder:** $85$

*(Verification: $Q \times 97 + 85 = 999,999,999,999,999,999,999,999,999,915 + 85 = 10^{30}$)*
