Casting out nines is a quick check on a multiplication, because a number and its digit sum leave the same remainder on division by 9. Only one of these can possibly be the value of $3456\times 789$. Which one?

The digits of 3456 add to 18, a multiple of 9, so 3456 is a multiple of 9 and the product must be one too. Now take the digit sums of the four candidates: 2725784 → 35, 2726884 → 37, 2736784 → 37 and 2726784 → 36. Only 36 is a multiple of 9, so only 2726784 can be right, and the long multiplication confirms $3456\times 789=2726784$. Remember the check only rules answers OUT — passing it is necessary but never proof on its own, since swapping two digits leaves the digit sum unchanged.