SMO Past year questions

Call this number $N$ . We need $N$ to divide $5$ , therefore the last digit must equal $0$ or $5$ . Furthermore, we need $N$ to divide $11$ , which means that the sum of the odd digits and the sum of the even digits must be equal mod $11$ .

The total sum of the digits is of course $0+1+2+3+4+5+6=21$ : an odd number, therefore the 'odd sum' (call it $O$ ) and 'even sum' (call it $E$ ) cannot be precisely equal. The only remaining possibility is that they differ by $11$ , meaning one is equal to $5$ and the other to $16$ . $O$ is the sum of $4$ different digits, meaning $O\geq 6$ . Therefore $O=16$ and $E=5$ .

To make $N$ as large as possible, we can take its first digit to be $6$ . The second digit cannot be equal to $5$ , seeing as it is one of three odd digits that sum to $5$ . We can take it to be $4$ , forcing the other odd digits to be $1$ and $0$ . Therefore the last digit must be equal to $5$ . Arranging the other digits from high to low (to make $N$ as large as possible), we obtain $N=\boxed{6431205}$ .