## Dividing 10^30 by 97

I need to find q and r such that 10^30 = 97q + r, where 0 ≤ r < 97.

### Finding the Remainder First

I need 10^30 mod 97.

Using Fermat's Little Theorem, since 97 is prime:
10^96 ≡ 1 (mod 97)

I need to find the order of 10 mod 97, or just compute 10^30 mod 97 directly.

**Building up powers of 10 mod 97:**

| Power | Value mod 97 |
|-------|-------------|
| 10^1 | 10 |
| 10^2 | 100 ≡ 3 |
| 10^3 | 30 |
| 10^4 | 300 ≡ 300 - 3(97) = 300 - 291 = 9 |
| 10^5 | 90 |
| 10^6 | 900 ≡ 900 - 9(97) = 900 - 873 = 27 |
| 10^7 | 270 ≡ 270 - 2(97) = 270 - 194 = 76 |
| 10^8 | 760 ≡ 760 - 7(97) = 760 - 679 = 81 |
| 10^9 | 810 ≡ 810 - 8(97) = 810 - 776 = 34 |
| 10^10 | 340 ≡ 340 - 3(97) = 340 - 291 = 49 |
| 10^15 | 10^10 · 10^5 ≡ 49 · 90 = 4410 ≡ 4410 - 45(97) = 4410 - 4365 = 45 |
| 10^20 | 10^10 · 10^10 ≡ 49 · 49 = 2401 ≡ 2401 - 24(97) = 2401 - 2328 = 73 |
| 10^30 | 10^20 · 10^10 ≡ 73 · 49 = 3577 ≡ 3577 - 36(97) = 3577 - 3492 = **85** |

### The Remainder is 85

### Finding the Quotient

$$q = \frac{10^{30} - 85}{97}$$

$$\boxed{q = 10{,}309{,}278{,}350{,}515{,}463{,}917{,}525{,}773{,}195{,}876}$$

**Verification:** 97 × 10,309,278,350,515,463,917,525,773,195,876 + 85 = 10^30 ✓

### Answer

$$10^{30} = 97 \times 10{,}309{,}278{,}350{,}515{,}463{,}917{,}525{,}773{,}195{,}876 + \mathbf{85}$$

- **Quotient:** 10,309,278,350,515,463,917,525,773,195,876
- **Remainder:** 85
