13782 ÷ 7 = 1968 R6
Alright, let's work through the division of 13782 by 7 step by step.

Step 1:
7 goes into 1 0 times with a remainder of 1.
Write down 0 as next digit of of the result. 
Result so far: 0
Subtract 0 from 1 to get 1.
Bring next digit (3) of the dividend behind the 1 and repeat the process: 13 / 7

Step 2:
7 goes into 13 1 times with a remainder of 6.
Write down 1 as next digit of of the result. 
Result so far: 1
Subtract 7 from 13 to get 6.
Bring next digit (7) of the dividend behind the 6 and repeat the process: 67 / 7

Step 3:
7 goes into 67 9 times with a remainder of 4.
Write down 9 as next digit of of the result. 
Result so far: 19
Subtract 63 from 67 to get 4.
Bring next digit (8) of the dividend behind the 4 and repeat the process: 48 / 7

Step 4:
7 goes into 48 6 times with a remainder of 6.
Write down 6 as next digit of of the result. 
Result so far: 196
Subtract 42 from 48 to get 6.
Bring next digit (2) of the dividend behind the 6 and repeat the process: 62 / 7

Step 5:
7 goes into 62 8 times with a remainder of 6.
Write down 8 as next digit of of the result. 
Result so far: 1968
Subtract 56 from 62 to get 6.

The final result is 1968 with a remainder of 6.