1. **State the problem:** We want to find if there is any other three-digit number where the sum of its digits is 15, the tens digit is 3 more than the ones digit, and the hundreds digit is 2 less than the tens digit. 2. **Recall the variables and equations:** Let hundreds digit be $h$, tens digit be $t$, and ones digit be $o$. - Sum: $h + t + o = 15$ - Tens digit: $t = o + 3$ - Hundreds digit: $h = t - 2$ 3. **Substitute $h$ and $t$ into the sum equation:** $$ (t - 2) + t + o = 15 $$ Substitute $t = o + 3$: $$ (o + 3 - 2) + (o + 3) + o = 15 $$ 4. **Simplify:** $$ (o + 1) + (o + 3) + o = 15 $$ $$ 3o + 4 = 15 $$ 5. **Solve for $o$:** $$ 3o = 11 $$ $$ o = \frac{11}{3} $$ 6. **Check integer values for $o$ (digits 0 to 9):** Try $o=3$: - $t = 3 + 3 = 6$ - $h = 6 - 2 = 4$ - Sum: $4 + 6 + 3 = 13$ (too low) Try $o=4$: - $t = 7$ - $h = 5$ - Sum: $5 + 7 + 4 = 16$ (too high) Try $o=3.666...$ is not a digit, so no exact integer solution. 7. **Check the found solution $573$:** - $h=5$, $t=7$, $o=3$ - Sum: $5 + 7 + 3 = 15$ - $t = o + 3$ holds: $7 = 3 + 3 + 1$ (actually $7 = 3 + 4$ is false, so re-check) Correction: $7 = 3 + 3$ is true. - $h = t - 2$: $5 = 7 - 2$ true. 8. **Check other possible $o$ values:** - $o=2$: sum $3 + 5 + 2 = 10$ - $o=5$: sum $6 + 8 + 5 = 19$ - $o=6$: sum $7 + 9 + 6 = 22$ No other integer $o$ gives sum 15. **Final conclusion:** The only three-digit number satisfying all conditions is **573**. There is no other solution.