The correct answer is option (C):
Both the statements together are needed to answer the question
Here's the breakdown of why the answer is "Both the statements together are needed to answer the question":
First, let's understand the divisibility rule for 132. Since 132 = 4 x 3 x 11, a number is divisible by 132 if and only if it's divisible by 4, 3, and 11. We need to check the conditions provided in each statement against these rules.
Statement 1:
* The last four digits of the number have a factor of 4. This implies the last two digits (EF) are divisible by 4. This is a condition for divisibility by 4, but only a necessary one; for divisibility by 4, the number must only have the last two digits divisible by 4.
* A + C + E = 12(B + D + F). This equation doesn't immediately tell us anything about divisibility by 3 or 11. While it can give some relationship between the sum of digits, this is not directly related to divisibility rules of 3 and 11. It is worth noting that if the number is divisible by 3, the sum of its digits would be divisible by 3. And if the sum of all digits is a multiple of 3, both sides are also divisible by 3.
Statement 2:
* The sum of all the digits (A + B + C + D + E + F) is divisible by 24. Since 24 = 3 x 8, and the sum of the digits must be divisible by 3. Because all digits sum up to 24, this guarantees that ABCDEF is divisible by 3. However, this doesn't directly tell us anything about divisibility by 4 or 11. It's related to the sum of the digits being divisible by 3, which is a condition for the number to be divisible by 3, but the 8 portion of the divisibility rule for 24 adds no value here.
Combining Statements 1 and 2:
* From Statement 1, we know the last two digits are a multiple of 4, thus ABCDEF is divisible by 4.
* From Statement 2, we know the sum of digits is divisible by 24, thus it must also be divisible by 3.
* We need to verify if ABCDEF is divisible by 11. A number is divisible by 11 if the alternating sum of its digits is divisible by 11. i.e., (A - B + C - D + E - F) is a multiple of 11.
* We know that A + C + E = 12(B + D + F). Therefore A+C+E is divisible by 3 and B+D+F is also divisible by 3.
* Since A + B + C + D + E + F is divisible by 24, (A + C + E) + (B + D + F) is divisible by 24.
* Let A + C + E = x and B + D + F = y. From statement 1, x = 12y. Hence 12y + y = 24k where k is an integer. Thus 13y = 24k, thus y = 24/13k, this would imply y is a multiple of 24. x is also a multiple of 24, as x = 12y
* Thus, we cannot say whether (A - B + C - D + E - F) is a multiple of 11. Therefore, even together, we cannot determine if the number is divisible by 11.
Because only statements 1 and 2 provide information about the number being divisible by 3 and 4 (conditions for 132), we can not be sure if the number is divisible by 11. Therefore, we need to combine both the statements to make a conclusion. We can conclude that neither statement alone is sufficient, however, both statements together do not answer the question.