Math Problem Statement
Solution
The question asks whether is an integer, given the following two statements:
- S1: is not an integer.
- S2: is an integer.
We need to determine whether the statements are sufficient to conclude that is an integer, using the following options:
- (a) S1 alone is sufficient.
- (b) S2 alone is sufficient.
- (c) Both S1 and S2 are required.
- (d) Both S1 and S2 are not sufficient.
Analysis:
-
S1: is not an integer
This tells us that is not divisible by 3. However, it doesn't provide enough information to confirm whether is an integer. For example, would satisfy not being an integer, but would also satisfy this, and is not an integer.
S1 alone is not sufficient. -
S2: is an integer
This means that is a rational number, because if is an integer, then could be any number that, when multiplied by 3, gives an integer. For example, would satisfy this, but so would , which is not an integer.
S2 alone is also not sufficient. -
Combining S1 and S2
If we know that is not an integer (S1) and that is an integer (S2), then cannot be a non-integer multiple of 3 (like ). Therefore, the only possible solution is that is an integer.
Both S1 and S2 together are sufficient to conclude that is an integer.
Correct Answer:
(c) Both required.
Would you like further details or have any questions?
Here are 5 related questions to extend the discussion:
- What happens if is an integer?
- Can you provide an example where holds but doesn't?
- How can you prove that a number is an integer from the given conditions?
- Is it possible for both and to be non-integers?
- Can this method be generalized to other divisibility problems?
Tip: In data sufficiency problems, always check if combining the statements provides additional insight that individual statements lack.
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Math Problem Analysis
Mathematical Concepts
Algebra
Divisibility
Rational Numbers
Integers
Formulas
Divisibility rules for integers
Theorems
Integer properties related to multiplication and division
Suitable Grade Level
Grades 9-12
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