1. **State the problem:** We need to find digits $x$ and $y$ such that the number $42x4y$ is divisible by 72. 2. **Understand divisibility rules:** 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. Here, the last three digits are $x4y$. 4. **Divisibility by 9 rule:** The sum of all digits must be divisible by 9. The digits are $4, 2, x, 4, y$. 5. **Apply divisibility by 9:** Sum of digits is $4 + 2 + x + 4 + y = 10 + x + y$. This sum must be divisible by 9. 6. **Apply divisibility by 8:** The number formed by the last three digits is $100x + 40 + y$. This must be divisible by 8. 7. **Check divisibility by 8:** Since $100 mod 8 = 4$, the expression modulo 8 is $4x + 40 + y mod 8 = 4x + y + 0 mod 8$ (because $40 mod 8 = 0$). So, $4x + y mod 8 = 0$. 8. **Try values for $x$ and $y$ (digits 0 to 9) to satisfy both conditions:** - From divisibility by 9: $10 + x + y$ divisible by 9. - From divisibility by 8: $4x + y$ divisible by 8. 9. **Test possible values:** - For $x=1$, $4(1) + y = 4 + y$ divisible by 8. Possible $y$ values: $4$ (since $4+4=8$), $12$ (not digit), so $y=4$. Check sum: $10 + 1 + 4 = 15$, not divisible by 9. - For $x=2$, $4(2) + y = 8 + y$ divisible by 8. Since 8 is divisible by 8, $y$ must be divisible by 8. Possible $y=0,8$. Check sums: $10 + 2 + 0 = 12$ not divisible by 9. $10 + 2 + 8 = 20$ not divisible by 9. - For $x=3$, $4(3) + y = 12 + y mod 8 = 4 + y mod 8 = 0$ so $y mod 8 = 4$. Possible $y=4$. Sum: $10 + 3 + 4 = 17$ not divisible by 9. - For $x=4$, $4(4) + y = 16 + y mod 8 = 0 + y mod 8 = 0$ so $y mod 8 = 0$. Possible $y=0,8$. Sum: $10 + 4 + 0 = 14$ no. $10 + 4 + 8 = 22$ no. - For $x=5$, $4(5) + y = 20 + y mod 8 = 4 + y mod 8 = 0$ so $y mod 8 = 4$. $y=4$. Sum: $10 + 5 + 4 = 19$ no. - For $x=6$, $4(6) + y = 24 + y mod 8 = 0 + y mod 8 = 0$ so $y mod 8 = 0$. $y=0,8$. Sum: $10 + 6 + 0 = 16$ no. $10 + 6 + 8 = 24$ yes, divisible by 9. 10. **Check final candidate:** $x=6$, $y=8$. Last three digits: $6 4 8 = 648$. Check divisibility by 8: $648 mod 8 = 0$ (since $8 imes 81 = 648$). Sum digits: $4 + 2 + 6 + 4 + 8 = 24$, divisible by 9. Therefore, $x=6$ and $y=8$. **Final answer:** $x=6$, $y=8$.