Math Problem Statement

Is x an integer? S1: x/3 is not an integer. S2: 3x is an integer.

Solution

The question asks whether xx is an integer, given the following two statements:

  • S1: x3\frac{x}{3} is not an integer.
  • S2: 3x3x is an integer.

We need to determine whether the statements are sufficient to conclude that xx 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:

  1. S1: x3\frac{x}{3} is not an integer
    This tells us that xx is not divisible by 3. However, it doesn't provide enough information to confirm whether xx is an integer. For example, x=5x = 5 would satisfy 53\frac{5}{3} not being an integer, but x=52x = \frac{5}{2} would also satisfy this, and 52\frac{5}{2} is not an integer.
    S1 alone is not sufficient.

  2. S2: 3x3x is an integer
    This means that xx is a rational number, because if 3x3x is an integer, then xx could be any number that, when multiplied by 3, gives an integer. For example, x=1x = 1 would satisfy this, but so would x=13x = \frac{1}{3}, which is not an integer.
    S2 alone is also not sufficient.

  3. Combining S1 and S2
    If we know that x3\frac{x}{3} is not an integer (S1) and that 3x3x is an integer (S2), then xx cannot be a non-integer multiple of 3 (like 13\frac{1}{3}). Therefore, the only possible solution is that xx is an integer.
    Both S1 and S2 together are sufficient to conclude that xx 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:

  1. What happens if x3\frac{x}{3} is an integer?
  2. Can you provide an example where S1S1 holds but S2S2 doesn't?
  3. How can you prove that a number is an integer from the given conditions?
  4. Is it possible for both x3\frac{x}{3} and 3x3x to be non-integers?
  5. 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