Let’s solve \( 10^{30} \div 97 \):

#### Step 1: Find the quotient

The *quotient* is the result of integer division; the *remainder* is what's left.

Here, we want:
\[
Q = \left\lfloor \frac{10^{30}}{97} \right\rfloor
\]
and the remainder \( r \) so that:
\[
10^{30} = 97Q + r,\, 0 \leq r < 97
\]

#### Step 2: Express the remainder \( r \)

The remainder is:
\[
r = 10^{30} \bmod 97
\]

Let's compute \( 10^{30} \bmod 97 \):

##### Use modular exponentiation

**Find the order of 10 mod 97:**
- \( 10^1 \bmod 97 = 10 \)
- \( 10^2 = 100 \bmod 97 = 3 \)
- \( 10^3 = 30 \)
- \( 10^4 = 9 \)
- \( 10^6 = (10^3)^2 = 900 \bmod 97 = 27 \)
- \( 10^{10} = (10^5)^2 \)
But let’s make it quick:

Let’s use repeated squaring:

- \( 10^1 = 10 \bmod 97 \)
- \( 10^2 = 100 = 3 \bmod 97 \)
- \( 10^4 = 3^2 = 9 \bmod 97 \)
- \( 10^8 = 9^2 = 81 \bmod 97 \)
- \( 10^{16} = 81^2 = 6561 \bmod 97 \)

\( 6561 \div 97 = 67.6289 \), \( 97 \times 67 = 6499 \), \( 6561-6499=62 \)
So \( 10^{16} \bmod 97 = 62 \)

Let's now multiply accordingly to get \( 10^{30} \):
- \( 30 = 16+8+4+2 \)

So:
\[
10^{30} = 10^{16} \times 10^8 \times 10^4 \times 10^2 \bmod 97
\]
Plug them in:
- \( 10^{16} \bmod 97 = 62 \)
- \( 10^8 \bmod 97 = 81 \)
- \( 10^4 \bmod 97 = 9 \)
- \( 10^2 \bmod 97 = 3 \)

Now multiply stepwise:
\[
(62 \times 81) \bmod 97
\]
\( 62 \times 81 = 5022 \)
\( 5022 \div 97 = 51.752 \)
\( 97 \times 51 = 4947 \)
\( 5022-4947=75 \)

So, \( 62 \times 81 \bmod 97 = 75 \)

\[
(75 \times 9) \bmod 97 = 675 \bmod 97
\]
\( 97 \times 6 = 582 \), \( 675-582=93 \)
So \( 75 \times 9 \bmod 97 = 93 \)

\[
93 \times 3 = 279 \), \( 97 \times 2 = 194 \), \( 279-194=85 \)

So \( 10^{30} \bmod 97 = \boxed{85} \)

#### Step 3: The quotient

\[
Q = \frac{10^{30} - 85}{97}
\]
So:
\[
Q = \left\lfloor \frac{10^{30}}{97} \right\rfloor
\]

#### **Final Answer**:

- **Quotient**: \( Q = \frac{10^{30} - 85}{97} \)
- **Remainder**: \( \boxed{85} \)

Or, in words:

> **When \( 10^{30} \) is divided by 97, the quotient is \( Q = (10^{30} - 85)/97 \), and the remainder is 85.**
