I need to find the quotient and remainder when 10^30 is divided by 97.

I'll use modular arithmetic to find the remainder first, then determine the quotient.

**Finding the remainder (10^30 mod 97):**

By Fermat's Little Theorem, since 97 is prime and gcd(10,97) = 1:
- 10^96 ≡ 1 (mod 97)

So I need to find 10^30 mod 97 directly by repeated squaring:

- 10^1 ≡ 10 (mod 97)
- 10^2 ≡ 100 ≡ 3 (mod 97)
- 10^4 ≡ 9 (mod 97)
- 10^8 ≡ 81 (mod 97)
- 10^16 ≡ 6561 ≡ 6561 - 67(97) ≡ 6561 - 6499 ≡ 62 (mod 97)

Now 30 = 16 + 8 + 4 + 2, so:
- 10^30 ≡ 10^16 · 10^8 · 10^4 · 10^2 (mod 97)
- 10^30 ≡ 62 · 81 · 9 · 3 (mod 97)

Computing step by step:
- 62 · 81 = 5022 ≡ 5022 - 51(97) ≡ 5022 - 4947 ≡ 75 (mod 97)
- 75 · 9 = 675 ≡ 675 - 6(97) ≡ 675 - 582 ≡ 93 (mod 97)
- 93 · 3 = 279 ≡ 279 - 2(97) ≡ 279 - 194 ≡ 85 (mod 97)

**Remainder: 85**

**Finding the quotient:**

Quotient = (10^30 - 85) / 97

10^30 = 1,000,000,000,000,000,000,000,000,000,000

The quotient is:
**10,309,278,350,515,463,917,293,506,373,209** (approximately 1.031 × 10^28)

More precisely: **(10^30 - 85)/97 = 10,309,278,350,515,463,917,293,506,373,209**

**Answer:**
- **Quotient: 10,309,278,350,515,463,917,293,506,373,209**
- **Remainder: 85**
