Okay, we are given 5782 and 5677. Let's add them up step by step.

Step 1: We'll start by adding the digits 2 & 7 in column 1 and get 9.

Step 2: We'll start by adding the digits 8 & 7 in column 2 and get 15.
We'll write down the last digit 5 and carry the 1 to the next column.

Step 3: We'll start by adding the digits 7 & 6 in column 3 and get 14.
We'll write down the last digit 4 and carry the 1 to the next column.

Step 4: We'll start by adding the digits 5 & 5 in column 4 and get 11.
We'll write down the last digit 1 and carry the 1 to the next column.

So, 5782 + 5677 = 11459.

