ABC is a three-digit number; when ABC is multiplied by a single-digit number…

Concept

When a known product P is written as (a three-digit number) × (a single-digit multiplier), that multiplier must be a single-digit divisor of P whose quotient has three digits. Pairing this with any constraint on the number's digits isolates one candidate; only then are the claimed properties tested on that candidate.

Application

1. The product is 2634, and it equals ABC × X with X a single digit (1 to 9).

2. Factorise to find single-digit divisors: 2634 = 2 × 3 × 439, so its single-digit divisors are 1, 2, 3 and 6.

3. Keep only divisors giving a three-digit quotient: X = 1 gives 2634 and X = 2 gives 1317 (both four-digit, rejected); X = 3 gives 878; X = 6 gives 439.

4. Apply “tens digit = X/2”: for X = 3 the tens digit would need to be 1.5 (impossible) and 878 has tens digit 7, so reject; for X = 6 the tens digit must be 3, and 439 has tens digit 3 — a match.

5. Hence ABC = 439 with X = 6 is the only number meeting every condition.

Cross-check the statements on 439

• Unit digit: 439 ends in 9, so statement (i) holds.

• Primality: since 202 = 400 < 439 < 441 = 212, only primes up to 20 need testing; none of 2, 3, 5, 7, 11, 13, 17, 19 divides 439, so 439 is prime and statement (ii) holds.

• Digit sum: 4 + 3 + 9 = 16, so statement (iii) holds.

All three statements are verified for 439, so the correct choice is the one naming statements (i), (ii) and (iii) together.

Cross-check the product

439 × 6 = 2634, which confirms the resolved number.