15665 ÷ 31 
Our goal is to divide 15665 by 31.

Step 1:
The number 31 fits into 1 0 times, leaving a remainder of 1.
Use 0 as the next digit of our solution. 
Result so far: 0.0
The remainder is 1 after subtracting 0 from 1.
Include the next digit (5) from the dividend after 1, then repeat: 15 / 31

Moving on to step 2:
If we divide 15 by 31, we get 0 and a remainder of 15.
Put 0 as the next digit of the answer. 
Result so far: 0.0
Subtract 0 from 15 to get 15.
Take the next digit (6) from the dividend and append it to 15, then repeat: 156 / 31

Advancing to step 3:
31 goes into 156 5 times with a remainder of 1.
The number 5 becomes the next digit in our result. 
Result so far: 5.0
The remainder is 1 after subtracting 155 from 156.
Take the next digit (6) from the dividend and append it to 1, then repeat: 16 / 31

On to step 4:
16 divided by 31 is 0 with a remainder of 16.
Use 0 as the next digit of our solution. 
Result so far: 50.0
The remainder is 16 after subtracting 0 from 16.
Include the next digit (5) from the dividend after 16, then repeat: 165 / 31

Advancing to step 5:
The number 31 fits into 165 5 times, leaving a remainder of 10.
Use 5 as the next digit of our solution. 
Result so far: 505.0
Subtracting 155 from 165 leaves us with 10.
Since there are no more digits in the dividend, we add a zero to the remainder making it 100, and continue the process.

Let's proceed to step 6:
31 can be fit into 100 3 times, resulting in a remainder of 7.
Record the quotient 3 as the next digit in the result. 
Result so far: 505.3
If we take 93 away from 100, we end up with 7.
Since there are no more digits in the dividend, we add a zero to the remainder making it 70, and continue the process.

Advancing to step 7:
When dividing 70 by 31, we get 2 with a remainder of 8.
Put 2 as the next digit of the answer. 
Result so far: 505.32
If we take 62 away from 70, we end up with 8.
Since there are no more digits in the dividend, we add a zero to the remainder making it 80, and continue the process.

Step 8:
When dividing 80 by 31, we get 2 with a remainder of 18.
Record the quotient 2 as the next digit in the result. 
Result so far: 505.322
The remainder is 18 after subtracting 62 from 80.
Since there are no more digits in the dividend, we add a zero to the remainder making it 180, and continue the process.

The final result is 505.322 with a remainder of 0.0018.