1. **Problem statement:** We need to find digits $x$ and $y$ such that the number $42x4y$ is divisible by 72. 2. **Key fact:** A number is divisible by 72 if and only if it is divisible by both 8 and 9. 3. **Divisibility by 8 rule:** The last three digits must be divisible by 8. 4. The last three digits are $x4y$. We consider the number $100x + 40 + y$. 5. **Divisibility by 9 rule:** The sum of all digits must be divisible by 9. 6. The sum of digits is $4 + 2 + x + 4 + y = 10 + x + y$. 7. **Check divisibility by 8:** We test values of $x$ and $y$ from 0 to 9 to find $100x + 40 + y$ divisible by 8. 8. For $x=1$, $100(1)+40+y=140+y$. Check $140+y$ mod 8: $140 \div 8 = 17$ remainder $4$, so $y$ must be $4$ to make remainder 0. So $y=4$ works for $x=1$. 9. **Check divisibility by 9:** Sum is $10 + 1 + 4 = 15$, which is not divisible by 9. 10. Try $x=3$: $100(3)+40+y=340+y$. $340 \div 8 = 42$ remainder 4, so $y=4$ again to get remainder 0. Sum of digits: $10 + 3 + 4 = 17$, not divisible by 9. 11. Try $x=5$: $540 + y$. $540 \div 8 = 67$ remainder 4, so $y=4$ again. Sum: $10 + 5 + 4 = 19$, no. 12. Try $x=7$: $740 + y$. $740 \div 8 = 92$ remainder 4, so $y=4$. Sum: $10 + 7 + 4 = 21$, divisible by 9? $21 \div 9 = 2$ remainder 3, no. 13. Try $x=9$: $940 + y$. $940 \div 8 = 117$ remainder 4, so $y=4$. Sum: $10 + 9 + 4 = 23$, no. 14. Try $x=0$: $40 + y$. $40 \div 8 = 5$ remainder 0, so $y$ can be 0. Sum: $10 + 0 + 0 = 10$, no. 15. Try $x=2$: $240 + y$. $240 \div 8 = 30$ remainder 0, so $y=0$. Sum: $10 + 2 + 0 = 12$, no. 16. Try $x=4$: $440 + y$. $440 \div 8 = 55$ remainder 0, so $y=0$. Sum: $10 + 4 + 0 = 14$, no. 17. Try $x=6$: $640 + y$. $640 \div 8 = 80$ remainder 0, so $y=0$. Sum: $10 + 6 + 0 = 16$, no. 18. Try $x=8$: $840 + y$. $840 \div 8 = 105$ remainder 0, so $y=0$. Sum: $10 + 8 + 0 = 18$, which is divisible by 9. 19. So the digits are $x=8$ and $y=0$. 20. **Final answer:** $x=8$, $y=0$. **Verification:** Number is $42840$. $42840 \div 8 = 5355$ (integer), and sum of digits $4+2+8+4+0=18$ divisible by 9. Hence, $42840$ is divisible by 72.