The correct answer is option (C):
both the statements together are needed to answer the question
Let's analyze the problem and the statements provided to determine if we can find the value of a x b x c, where a, b, and c are prime numbers.
Statement 1: a + b + c = 12
This statement alone is insufficient. We know the sum of three prime numbers is 12, but there are multiple combinations of prime numbers that add up to 12. For example, 2 + 3 + 7 = 12 and 2 + 2 + 8 (8 is not prime) are possibilities. However, since the prime numbers must be integers, the primes adding up to 12 must be 2, 3 and 7. Thus, a x b x c would be 2 x 3 x 7 = 42. Since we found a unique answer, the question can be answered by statement 1.
Statement 2: 6300 is divisible by a x b x c and 1890 is divisible by b x c
This statement alone is also insufficient. We know that a x b x c is a factor of 6300 and b x c is a factor of 1890. Let's find the prime factorizations:
6300 = 2^2 x 3^2 x 5^2 x 7
1890 = 2 x 3^3 x 5 x 7
From the factorization of 1890, we can say that b and c are factors of 2 x 3^3 x 5 x 7. From the factorization of 6300, a, b, and c can be found.
However, we can't determine the exact value of a, b, and c from this statement alone. We only know they are primes and they are factors of 6300 and the product of b and c is a factor of 1890.
Combining both statements:
Statement 1 gives us a + b + c = 12, implying the prime numbers are 2, 3, and 7. Therefore, a x b x c = 2 x 3 x 7 = 42.
Statement 2 tells us that a, b, and c are factors of 6300 and b and c are factors of 1890. The prime factorization of 6300 is 2^2 x 3^2 x 5^2 x 7 and 1890 is 2 x 3^3 x 5 x 7. Therefore, the prime factors must be within these factorizations.
Since the only primes that sum to 12 are 2, 3, and 7 (from Statement 1), we can determine that a, b, and c must be 2, 3, and 7. These numbers also fit the criteria outlined by Statement 2 (they are factors of 6300 and the product of any two of them, e.g., 3 and 7 is a factor of 1890). Therefore, the value of a x b x c = 2 x 3 x 7 = 42
Since we need both statements to find the values of a, b and c, the correct answer is: "both the statements together are needed to answer the question".