Question

S is a set of consecutive numbers. What is the size of the set?
i) There are two multiples of 5 in the set.
ii) There are seven multiples of 2 in the set.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. EACH statement ALONE is sufficient.
• D. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

For problems about sets of consecutive integers, the range of possible sizes depends on the start and end points relative to the multiples. A set of size \(N\) has either \(\lfloor N/k \rfloor\) or \(\lceil N/k \rceil\) multiples of \(k\). Always test edge cases (e.g., starting or ending on a multiple) to check for ambiguity.

Answer 1

Answer: (E)

Step 1: Understanding the Concept
The size of a set of consecutive integers is given by (Last term - First term + 1). The number of multiples of 'n' in such a set depends on the size of the set and its starting point. We need to determine if the given statements can uniquely determine the size of the set S.
Step 2: Analyze Statement (1)
"There are two multiples of 5 in the set."
Let the two multiples be \(5k\) and \(5(k+1)\). The distance between them is 5.

Minimum size: The set could start just after a multiple of 5 and end exactly on the second one. E.g., S = \{6, 7, 8, 9, 10\}. Here, the size is 5, but there is only one multiple of 5. For two multiples, the set could be S = \{5, 6, 7, 8, 9, 10\}. Size = 6. Multiples are 5, 10.
Maximum size: The set could start just after a multiple of 5, include two more multiples, and end just before the next one. E.g., S = \{6, 7, ..., 14\}. Multiples are 10, (none). Example with two: S = \{1, 2, ..., 9\}. Multiples are 5. S = \{1, 2, ..., 10\}. Multiples are 5, 10.
Let's consider a set with two multiples of 5, say 5 and 10. The set could be \{5, 6, 7, 8, 9, 10\}, with a size of 6. Or it could be \{1, 2, ..., 10\}, with a size of 10.
The size of the set can range from 6 to 10, inclusive. Since the size is not unique, Statement (1) is NOT sufficient.
Step 3: Analyze Statement (2)
"There are seven multiples of 2 in the set."
Let the seven multiples be \(2k, 2(k+1), \dots, 2(k+6)\). The distance between the first and last is \(2(k+6) - 2k = 12\).

Minimum size: The set could be \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}. It contains multiples 2, 4, 6, 8, 10, 12, 14 (seven of them). The size is 13.
Maximum size: The set could be \{1, 2, ..., 14\}. It contains multiples 2, 4, 6, 8, 10, 12, 14. The size is 14.
The size of the set could be 13 or 14. Since the size is not unique, Statement (2) is NOT sufficient.
Step 4: Combine Statements (1) and (2)
We need a set of consecutive integers that contains exactly two multiples of 5 and exactly seven multiples of 2. From statement (2), the size is either 13 or 14.

Case A: Size = 13. Can a set of 13 consecutive integers have two multiples of 5 and seven multiples of 2? Let's try S = \{1, 2, ..., 13\}. Multiples of 2: 2, 4, 6, 8, 10, 12 (six). Fails. Let's try S = \{2, 3, ..., 14\}. Multiples of 2: 2, 4, 6, 8, 10, 12, 14 (seven). Multiples of 5: 5, 10 (two). This works. So, a size of 13 is possible.
Case B: Size = 14. Can a set of 14 consecutive integers have two multiples of 5 and seven multiples of 2? Let's try S = \{1, 2, ..., 14\}. Multiples of 2: 2, 4, 6, 8, 10, 12, 14 (seven). Multiples of 5: 5, 10 (two). This works. So, a size of 14 is possible.
Since a set of size 13 (e.g., \{2, ..., 14\}) and a set of size 14 (e.g., \{1, ..., 14\}) both satisfy the conditions, we still cannot determine a unique size for the set.
Step 5: Final Answer
Statements (1) and (2) together are NOT sufficient.