1. **State the problem:** Find a three-digit secret number below 750 that satisfies the following clues: - The ones digit divides evenly into each of the other two digits. - The number is even. - The sum of the ones digit and the tens digit equals the hundreds digit. - No two digits are the same. 2. **Analyze the clues:** - Let the number be $\overline{HTO}$ where $H$ is hundreds, $T$ is tens, and $O$ is ones digit. - Since the number is below 750, $H < 7.5$, so $H$ can be from 1 to 7. - The number is even, so $O$ (ones digit) is even: $O \in \{0,2,4,6,8\}$. - $O$ divides evenly into $H$ and $T$, so $H \bmod O = 0$ and $T \bmod O = 0$. - Sum condition: $H = T + O$. - All digits distinct. 3. **Check possible values for $O$:** - $O=0$ is invalid because division by zero is undefined. - $O=2$: - $H$ divisible by 2, so $H \in \{2,4,6\}$. - $T$ divisible by 2, so $T \in \{0,2,4,6,8\}$. - $H = T + 2$. Try $H=2$: - $2 = T + 2 \Rightarrow T=0$. - Check digits distinct: $H=2$, $T=0$, $O=2$ (digits 2,0,2) repeated 2, invalid. Try $H=4$: - $4 = T + 2 \Rightarrow T=2$. - Digits: 4,2,2 repeated 2, invalid. Try $H=6$: - $6 = T + 2 \Rightarrow T=4$. - Digits: 6,4,2 all distinct. - Check divisibility: 6 mod 2=0, 4 mod 2=0, valid. - Number is 642. - $O=4$: - $H$ divisible by 4: $H \in \{4\}$ (since $H$ is single digit and less than 7.5). - $T$ divisible by 4: $T \in \{0,4,8\}$. - $H = T + 4$. Try $H=4$: - $4 = T + 4 \Rightarrow T=0$. - Digits: 4,0,4 repeated 4, invalid. - $O=6$: - $H$ divisible by 6: $H=6$. - $T$ divisible by 6: $T=0,6$ (6 is single digit). - $H = T + 6$. Try $H=6$: - $6 = T + 6 \Rightarrow T=0$. - Digits: 6,0,6 repeated 6, invalid. - $O=8$: - $H$ divisible by 8: $H=8$ (not less than 7.5), invalid. 4. **Conclusion:** The only valid number is $642$. **Final answer:** $642$