Directions: Read the table carefully and answer the following question. The table shows the data about distance (km) covered from one point to another point by the vehicle used between each point and speed of each vehicle (km/h). Note: (i) If one person wants to travel from P to S, then he/she can only use a vehicle available at the point P. (ii) ** means there is no direct connection between given points. | | P | Q | R | S | Transport Vehicle | Speed (km/h) | |---|---|---|---|---|---|---| | P | 0 | 360 | ** | 780 | Train | 60 | | Q | 540 | 0 | ** | 600 | Taxi | 50 | | R | 420 | 720 | 0 | ** | Bus | 40 | | S | 630 | ** | 900 | 0 | Bike | 45 | A three-digit number PQR, when multiplied by a single digit number X, then the product is 2634. In the number PQR, the digit at tens place is equal to half of X. Which of the following statements is/are true about PQR? (i) Unit digit is 9 (ii) It is a prime number (iii) Sum of the digits is 18

### Concept & Factorization This problem requires reverse-engineering a multiplication equation by testing single-digit factors of a given product. $$ PQR \times X = 2634 $$ ### Step-by-Step Solution 1. **Find single-digit factors of 2634:** We need $X$ to be a single digit ($1 \le X \le 9$) such that dividing 2634 by $X$ results in a three-digit integer ($PQR$). - If $X = 2$: $2634 / 2 = 1317$ (This is a 4-digit number, so $X \neq 2$) - If $X = 3$: $2634 / 3 = 878$. Check condition: Is the tens digit (7) equal to half of $X$ (1.5)? No. - If $X = 6$: $2634 / 6 = 439$. Check condition: Is the tens digit (3) equal to half of $X$ (6 / 2 = 3)? Yes! 2. We have identified the numbers: $PQR = 439$ and $X = 6$. 3. **Evaluate the given statements about 439:** - **(i) Unit digit is 9:** The unit digit of 439 is indeed 9. (True) - **(ii) It is a prime number:** The square root of 439 is approximately 20.9. Testing prime factors up to 19 (2, 3, 5, 7, 11, 13, 17, 19): 439 is not divisible by any of them. Thus, it is prime. (True) - **(iii) Sum of the digits is 18:** $4 + 3 + 9 = 16 \neq 18$. (False) 4. Therefore, only statements (i) and (ii) are true. ### Exam Strategy & Shortcut To quickly find single-digit factors of 2634, use divisibility rules. The number ends in 4 (even, so divisible by 2). Sum of digits $2+6+3+4 = 15$ (divisible by 3). Since it is divisible by both 2 and 3, it is definitely divisible by 6. Testing 6 first directly gives you the correct framework. ### Common Pitfall Failing to verify prime numbers correctly. Remember you only need to test divisibility by prime numbers up to the square root of the target number to definitively prove it is prime. ### Final Answer Therefore, the correct answer is **Only (i) and (ii)**.