Okay, let's tackle this math problem. We're starting with 582435 and subtracting 66661.

Step 1: We'll start by subtracting the digit 1 and the borrow 0 from 5 in column 1 and get 4.
4 is the first digit of our result.

Step 2: We'll start by subtracting the digit 6 and the borrow 0 from 3 in column 2 and get -3.
We add -3 to 10 and get 7 as the next digit of the result.

Step 3: We'll start by subtracting the digit 6 and the borrow 1 from 4 in column 3 and get -3.
We add -3 to 10 and get 7 as the next digit of the result.

Step 4: We'll start by subtracting the digit 6 and the borrow 1 from 2 in column 4 and get -5.
We add -5 to 10 and get 5 as the next digit of the result.

Step 5: We'll start by subtracting the digit 6 and the borrow 1 from 8 in column 5 and get 1.
1 is the next digit of our result.

Step 6: We'll start by subtracting the digit 0 and the borrow 0 from 5 in column 6 and get 5.
5 is the next digit of our result.

So, 582435 - 66661 = 515774.