2562 ÷ 2 = 1281 R0
Alright, let's work through the division of 2562 by 2 step by step.

Step 1:
2 goes into 2 1 times with a remainder of 0.
Write down 1 as next digit of of the result. 
Result so far: 1
Subtract 2 from 2 to get 0.
Bring next digit (5) of the dividend behind the 0 and repeat the process: 5 / 2

Step 2:
2 goes into 5 2 times with a remainder of 1.
Write down 2 as next digit of of the result. 
Result so far: 12
Subtract 4 from 5 to get 1.
Bring next digit (6) of the dividend behind the 1 and repeat the process: 16 / 2

Step 3:
2 goes into 16 8 times with a remainder of 0.
Write down 8 as next digit of of the result. 
Result so far: 128
Subtract 16 from 16 to get 0.
Bring next digit (2) of the dividend behind the 0 and repeat the process: 2 / 2

Step 4:
2 goes into 2 1 times with a remainder of 0.
Write down 1 as next digit of of the result. 
Result so far: 1281
Subtract 2 from 2 to get 0.

The final result is 1281 with a remainder of 0.