Finding the Number for Divisibility

The problem asks for the smallest number, let's call it \(x\), that needs to be added to \(56789\) so that the sum \((56789 + x)\) is perfectly divisible by \(345\). After finding \(x\), we need to calculate the sum of its digits.

Calculate Remainder of 56789 divided by 345

To find \(x\), we first determine the remainder when \(56789\) is divided by \(345\).

Performing the division:

\( 56789 \div 345 \)

We find:

\( 56789 = 345 \times 164 + 209 \)

The remainder is \(R = 209\).

Determine the Smallest Number 'x'

For \(56789 + x\) to be divisible by \(345\), the term \((R + x)\) must be a multiple of \(345\). Since \(56789 = 345 \times 164 + 209\), we have:

\( 56789 + x = (345 \times 164 + 209) + x \)

We want this sum to be the next multiple of \(345\). Since the current remainder (\(209\)) is less than \(345\), the smallest multiple of \(345\) that is greater than \(209\) is \(345\) itself.

Therefore, we need:

\( 209 + x = 345 \)

Solving for \(x\):

\( x = 345 - 209 \)

\( x = 136 \)

The smallest number to add is \(x = 136\).

Calculate the Sum of Digits of x

The value of \(x\) is \(136\). We now find the sum of its digits.

Sum of digits of \(x = 1 + 3 + 6\)

\( \text{Sum} = 10 \)

The sum of the digits of \(x\) is \(10\).