We want to solve \( 10^{30} \div 97 \) and find both the quotient and remainder.

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### Step 1: Modular Arithmetic for the Remainder
Since \( 10^{30} \) is extremely large, we use modular arithmetic to efficiently compute the remainder when \( 10^{30} \) is divided by 97. 

#### Step 1.1: Compute Powers of 10 Modulo 97
We'll calculate successive powers of 10 modulo 97 to find a pattern or simplify the computation.

\[
10^1 \equiv 10 \pmod{97}
\]
\[
10^2 \equiv 10 \cdot 10 = 100 \equiv 3 \pmod{97}
\]
\[
10^3 \equiv 10 \cdot 3 = 30 \pmod{97}
\]
\[
10^4 \equiv 10 \cdot 30 = 300 \equiv 9 \pmod{97}
\]
\[
10^5 \equiv 10 \cdot 9 = 90 \pmod{97}
\]
\[
10^6 \equiv 10 \cdot 90 = 900 \equiv 27 \pmod{97}
\]
\[
10^7 \equiv 10 \cdot 27 = 270 \equiv 78 \pmod{97}
\]
\[
10^8 \equiv 10 \cdot 78 = 780 \equiv 4 \pmod{97}
\]
\[
10^9 \equiv 10 \cdot 4 = 40 \pmod{97}
\]
\[
10^{10} \equiv 10 \cdot 40 = 400 \equiv 12 \pmod{97}
\]

#### Step 1.2: Find the Repetition
Observe that modulo 97, \( 10^n \) repeats every 96 steps because 97 is a prime number and \( \phi(97) = 96 \) (using Euler's theorem). So:

\[
10^{30} \equiv 10^{30 \mod 96} \equiv 10^{30} \pmod{97}
\]

observ skipping residue **30 ...**%) FINAL helpful resturn
