1. **Problem statement:** We need to find the digit $P$ in a five-digit number $PQRST$ where each digit is from 1 to 5, used exactly once, and $P < Q$. 2. **Given conditions:** - The three-digit number $PQR$ is divisible by 2. - The three-digit number $QRS$ is divisible by 5. - The three-digit number $RST$ is divisible by 3. 3. **Important rules:** - Divisible by 2 means the last digit is even. - Divisible by 5 means the last digit is 0 or 5. Since digits are 1 to 5, last digit must be 5. - Divisible by 3 means the sum of digits is divisible by 3. 4. **Analyze divisibility by 5:** Since $QRS$ ends with $S$ and must be divisible by 5, $S=5$. 5. **Analyze divisibility by 2:** $PQR$ ends with $R$ and must be divisible by 2, so $R$ is even. Possible even digits are 2 and 4. 6. **Analyze divisibility by 3:** $RST$ must be divisible by 3, so sum $R + S + T$ is divisible by 3. We know $S=5$, so $R + 5 + T$ divisible by 3. 7. **Digits used:** Digits are 1, 2, 3, 4, 5 used once each. We have $S=5$, $R$ is 2 or 4. 8. **Try $R=2$:** Then $R+S+T = 2 + 5 + T = 7 + T$ divisible by 3. Possible $T$ values (from remaining digits excluding $R=2$ and $S=5$) are 1,3,4. - For $T=1$, sum = 8 (not divisible by 3). - For $T=3$, sum = 10 (not divisible by 3). - For $T=4$, sum = 11 (not divisible by 3). No valid $T$. 9. **Try $R=4$:** Then $R+S+T = 4 + 5 + T = 9 + T$ divisible by 3. Possible $T$ values are 1,2,3. - For $T=1$, sum = 10 (not divisible by 3). - For $T=2$, sum = 11 (not divisible by 3). - For $T=3$, sum = 12 (divisible by 3). So $T=3$. 10. **Digits used now:** $R=4$, $S=5$, $T=3$. Remaining digits for $P$ and $Q$ are 1 and 2. 11. **Condition $P < Q$:** Possible pairs: $(P,Q) = (1,2)$ or $(2,1)$ but $P < Q$ so $(1,2)$. 12. **Check $PQR$ divisibility by 2:** $PQR = 1 2 4$ ends with 4 (even), divisible by 2. 13. **Check $QRS$ divisibility by 5:** $QRS = 2 4 5$ ends with 5, divisible by 5. 14. **Check $RST$ divisibility by 3:** $R S T = 4 5 3$, sum = 12 divisible by 3. All conditions satisfied. **Answer:** $P = 1$. **Final answer:** C. 1