We can solve this together! We're beginning with 244034 and removing 66257 from it.

Step 1: We'll start by subtracting the digit 7 and the borrow 0 from 4 in column 1 and get -3.
We add -3 to 10 and get 7 as the first digit of the result.

Step 2: We'll start by subtracting the digit 5 and the borrow 1 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 2 and the borrow 1 from 0 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 4 in column 4 and get -3.
We add -3 to 10 and get 7 as the next digit of the result.

Step 5: We'll start by subtracting the digit 6 and the borrow 1 from 4 in column 5 and get -3.
We add -3 to 10 and get 7 as the next digit of the result.

Step 6: We'll start by subtracting the digit 0 and the borrow 1 from 2 in column 6 and get 1.
1 is the next digit of our result.

So, 244034 - 66257 = 177777.