Find the unit digit of $3^{65}\times 7^{42}\times 9^{27}$.

Unit digits cycle in 4s for 3 and 7 and in 2s for 9. $3^{65}:65=4\times 16+1$ → 3. $7^{42}:42=4\times 10+2$ → 9. $9^{27}$: odd power → 9. Product of unit digits $3\times 9\times 9=243$ → 3.