ABC is a three-digit number, when ABC is multiplied by a single digit number X, then the product is 2634. In that three-digit number ABC, the digit at tens place is equal to half of X. Which of the following statement/s is/are correct about ABC?

(i) Unit digit in ABC is 9
(ii) ABC is a prime number
(iii) Sum of digits of ABC is 16

Question

ABC is a three-digit number, when ABC is multiplied by a single digit number X, then the product is 2634. In that three-digit number ABC, the digit at tens place is equal to half of X. Which of the following statement/s is/are correct about ABC?

(i) Unit digit in ABC is 9  
(ii) ABC is a prime number  
(iii) Sum of digits of ABC is 16

Accepted Answer

**Solution:**

1. Given that $ABC 	imes X = 2634$, where $ABC$ is a three-digit number and $X$ is a single digit number.

2. Since $2634$ is the product, possible values of $X$ are factors of $2634$ that are single digit numbers. Checking single digit factors:
   - $2634 \div 3 = 878$ (not three-digit number with tens digit half of 3)
   - $2634 \div 6 = 439$

3. Check if $X=3$ satisfies the condition that the tens digit of $ABC$ is half of $X$:
   - Half of 3 is 1.5, which is not an integer digit, so $X=3$ is invalid.

4. Check for $X=6$:
   - Half of 6 is 3, so the tens digit of $ABC$ must be 3.
   - $ABC = 439$ (from $2634 \div 6$)
   - Tens digit of 439 is 3, which matches the condition.

5. Verify the statements:
   - (i) Unit digit in $ABC$ is 9: True (units digit of 439 is 9)
   - (ii) $ABC$ is a prime number: Check if 439 is prime.
     - 439 is not divisible by 2, 3, 5, 7, 11, 13, 17, or 19 (all primes less than $\sqrt{439} \approx 20.9$).
     - Hence, 439 is prime.
   - (iii) Sum of digits of $ABC$ is 16:
     - $4 + 3 + 9 = 16$

**Answer:** (d) All (i), (ii) and (iii)

Answer

Answer

Answer

Answer