Question

a, b and c are prime numbers. What is the value of a x b x c?
Statement 1: a + b + c = 12
Statement 2: 6300 is divisible by a x b x c and 1890 is divisible by b x c

• A. statement (1) alone is sufficient to answer the question
• B. statement (2) alone is sufficient to answer the question
• C. both the statements together are needed to answer the question
• D. statement (1) alone or statement (2) alone is sufficient to answer the question
• E. neither statement (1) nor statement (2) suffices to answer the question

Answer 1

The Correct Option is C

Step 1: Understand the problem.
We are asked to find the value of \( a \times b \times c \), where \( a \), \( b \), and \( c \) are prime numbers. The problem gives two statements:
- Statement 1: \( a + b + c = 12 \)
- Statement 2: 6300 is divisible by \( a \times b \times c \) and 1890 is divisible by \( b \times c \).

Step 2: Analyze Statement 1.
Statement 1 tells us that the sum of the three primes \( a + b + c = 12 \). The prime numbers that add up to 12 are:
\( a = 2 \), \( b = 3 \), and \( c = 7 \), because \( 2 + 3 + 7 = 12 \).
Therefore, from Statement 1, we know the values of \( a \), \( b \), and \( c \): \( a = 2 \), \( b = 3 \), and \( c = 7 \). The product of these primes is:
\[ a \times b \times c = 2 \times 3 \times 7 = 42 \] So, from Statement 1, we can directly conclude that \( a \times b \times c = 42 \).

Step 3: Analyze Statement 2.
Statement 2 tells us that 6300 is divisible by \( a \times b \times c \) and 1890 is divisible by \( b \times c \). Let's verify the divisibility conditions:
- \( 6300 \div 42 = 150 \), so 6300 is divisible by 42.
- \( 1890 \div 21 = 90 \), so 1890 is divisible by \( b \times c = 3 \times 7 = 21 \).

This confirms that \( a \times b \times c = 42 \) is correct.

Step 4: Conclusion.
Statement 1 alone gives the value of \( a \times b \times c \), so we don't need Statement 2 to answer the question. However, Statement 2 also confirms the divisibility conditions, supporting the answer.

Final Answer:
The correct option is (C): both the statements together are needed to answer the question.