Determine if 1781 is divisible by 3, 6, 7, 8, and 9

Let us determine if 1781 is divisible by 3, 6, 7, 8, and 9.

Since the last digit 1 is odd, we conclude that 1781 is not divisible by 2, and hence 1781 is not divisible by $6=2\cdot 3$ and 1781 is not divisible by $8=2\cdot 4$.

Since the sum $1+7+8+1=17$ of digits is not divisible by 3 (and hence by 9), we conclude that 1781 is not divisible by $3$ and 1781 is not divisible by $9$.

Taking into account that $178-2\cdot 1=176$, $17-2\cdot 6=5$, and $5$ is not divisible by 7, we conclude that 1781 is not divisible by 7.