Solution 1: 123450789 leaves a remainder of 7 when divided by 91, and 1000 leaves a remainder of 90, or -1, so adding 7 multiples of 1000 will give us a multiple of 91. Solution 2: For those who don't like long division, there is a quicker way. First notice that $91=7 \cdot 13$, and $7 \cdot 11 \cdot 13=1001$. Observe that $12345 n 789=123 \cdot 1001000+45 n \cdot 1001-123 \cdot 1001+123-45 n+789$. It follows that 91 will divide $12345 n 789$ iff 91 divides $123-45 n+789=462-n$. The number 462 is divisible by 7 and leaves a remainder of $\mathbf{7}$ when divided by 13.