I need to divide 10^30 by 97 and find both the quotient and remainder.

To find the remainder of 10^30 ÷ 97, I'll use modular arithmetic. First, I'll find a pattern in powers of 10 modulo 97.

Let me calculate the first few powers of 10 modulo 97:
- 10^1 ≡ 10 (mod 97)
- 10^2 ≡ 100 ≡ 3 (mod 97)
- 10^3 ≡ 30 (mod 97)
- 10^4 ≡ 300 ≡ 9 (mod 97)
- 10^5 ≡ 90 (mod 97)
- 10^6 ≡ 900 ≡ 27 (mod 97)

I'll use Fermat's Little Theorem since 97 is prime. By Fermat's Little Theorem:
10^96 ≡ 1 (mod 97)

Now I need to express 30 in terms of 96:
30 = 0 × 96 + 30

Therefore:
10^30 ≡ 10^30 (mod 97)

I'll calculate this by repeated squaring:
- 10^1 ≡ 10 (mod 97)
- 10^2 ≡ 3 (mod 97)
- 10^4 ≡ 9 (mod 97)
- 10^8 ≡ 81 (mod 97)
- 10^16 ≡ 81^2 = 6561 ≡ 67 × 97 + 62 ≡ 62 (mod 97)

Now: 30 = 16 + 8 + 4 + 2

So: 10^30 ≡ 10^16 × 10^8 × 10^4 × 10^2 ≡ 62 × 81 × 9 × 3 (mod 97)

Calculating step by step:
- 62 × 81 = 5022 ≡ 51 × 97 + 75 ≡ 75 (mod 97)
- 75 × 9 = 675 ≡ 6 × 97 + 93 ≡ 93 (mod 97)
- 93 × 3 = 279 ≡ 2 × 97 + 85 ≡ 85 (mod 97)

Therefore, the remainder is 85.

To find the quotient:
10^30 = 97q + 85

So: q = (10^30 - 85) ÷ 97

The quotient is:
q = 10309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567

Therefore:
- Quotient = 10309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567
- Remainder = 85
