To find the smallest number that should be added to 16809 so that the sum is completely divisible by 41, we follow these steps:

1. Divide the number by 41

Perform the division of 16809 by 41 to find the remainder:

$$16809 \div 41$$

$41 \times 400 = 16400$

$16809 - 16400 = 409$

$41 \times 9 = 369$

$409 - 369 = 40$

So, $16809 = (41 \times 409) + 40$.

The remainder is 40.

2. Calculate the number to be added

To make the number perfectly divisible by 41, the remainder needs to reach the next multiple of 41.

$$\text{Number to be added} = \text{Divisor} - \text{Remainder}$$

$$\text{Number to be added} = 41 - 40 = 1$$

Verification

If we add 1 to 16809, we get 16810.

$$16810 \div 41 = 410$$

Since 16810 is perfectly divisible by 41, the smallest number to be added is 1.