Question

A, B, C, D, E and F are six whole numbers. Is "ABCDEF" divisible by 132?
Statement 1: The last four digits of the given number has a factor 4 and A+ C+E = 12(B + D + F)
Statement 2: Sum of all the digits of the given number is divisible by 24

• 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. Either 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

To determine if the number "ABCDEF" is divisible by 132, we will evaluate the divisibility using the given statements. A number is divisible by 132 if it meets the criteria for divisibility by both 4 and 33. Here, 33 is a combination of 3 and 11. Therefore, we need to check:

• Divisibility by 4
• Divisibility by 3
• Divisibility by 11

Let's start by analyzing each statement:

Statement 1:

The last four digits of the number have a factor 4, and A + C + E = 12(B + D + F).

• For divisibility by 4, the last two digits of a number must be divisible by 4. The statement tells us that the last four digits have a factor 4, implying that the number ending in the last two digits is completely divisible by 4.
• The equation A + C + E = 12(B + D + F) provides a relationship but does not directly tell us about the divisibility by 3 or 11.

Statement 2:

The sum of all the digits of the number is divisible by 24.

• For a number to be divisible by 3, the sum of its digits must be divisible by 3. If the sum of all digits is divisible by 24, it is also divisible by 3.
• This statement does not provide direct information about divisibility by 4 or 11.

Combining Both Statements:

When these two statements are combined:

• From Statement 1, we know the number is divisible by 4.
• From Statement 2, we confirm divisibility by 3, as the sum of digits is divisible by 24 (which covers divisibility by 3).
• Neither statement separately provides information on divisibility by 11. However, knowing the sum of the parts (A + C + E and B + D + F as divisible by 24) and the specific equation might assist in verifying divisibility by 11 upon rewriting the digits to balance out the alternating sum of digits.

Hence, utilizing both statements together confirms divisibility by 132, since conditions for all components (4, 3, and potentially verified divisibility by 11) are satisfied as per requirements from these two statements.

Therefore, the correct answer is: Both the statements together are needed to answer the question.