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

First, let me calculate 10^30 ÷ 97.

To find the quotient and remainder, I'll use the division algorithm: 10^30 = 97q + r, where 0 ≤ r < 97.

Let me calculate 10^30 mod 97 first to find the remainder.

Using Fermat's Little Theorem: Since 97 is prime and gcd(10, 97) = 1, we have:
10^96 ≡ 1 (mod 97)

So I need to find 10^30 mod 97.

Let me calculate powers of 10 mod 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)

Let me use repeated squaring:
- 10^2 ≡ 3 (mod 97)
- 10^4 ≡ 9 (mod 97)
- 10^8 ≡ 81 (mod 97)
- 10^16 ≡ 81^2 ≡ 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)

Let me calculate 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)

So the remainder is **85**.

The quotient is:
q = (10^30 - 85)/97

q = 10309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567

**Quotient:** 10309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567

**Remainder:** 85
