The GMAT DS question given below is a data sufficiency question from Number Theory. Concept tested: Test of divisibility. Level of difficulty: Easy | A GMAT 600 to 650 level question.
This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -
All numbers used are real numbers.
A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)
Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight
You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.
All figures lie in a plane unless otherwise indicated.
In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.
Question 27: Is the positive integer 'x' divisible by 12?
What kind of an answer will the question fetch?
The question is an "Is" question. Answer to an "is" questions is either YES or NO.
When is the data sufficient?
The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.
What is the test of divisibility for 12?
The test of divisibility for 12 is that the number should be divisible by both 3 and 4. Essentially, x should be divisible by 3 and 22.
Approach: Look for a counter example
Example: x = 6. It is divisible by 6. However, it is NOT divisible by 12.
Counter Example: x = 12. It is divisible by 6. It is divisible by 12 as well.
Knowing that x is divisible by 6 is not enough to answer the question.
If x is divisible by 6, we can infer that it is divisible by 3 and 2. But we cannot deduce whether it is also divisible by 22 - which is essential to deduce that x is divisible by 12.
Statement 1 ALONE is NOT sufficient.
Eliminate choices A and D. Choices narrow down to B, C, or E.
If x is divisible by 8, then x will definitely be divisible by 4.
However, from statement (2) alone we do not know if x is divisible by 3.
Alternative Approach: Look for a counter example
Example: x = 8. It is divisible by 8. However, it is NOT divisible by 12.
Counter Example:x = 24. It is divisible by 8. It is divisible by 12 as well.
Knowing that x is divisible by 8 is not enough to answer the question.
Statement 2 ALONE is NOT sufficient.
Eliminate choice B. Choices narrow down to C or E.
From statement 1, if x is divisible by 6, it is definitely divisible by 3.
From statement 2, if x is divisible by 8, it is definitely divisible by 4.
So, by combining the two statements, we can conclude that x is divisible by 3 and by 4.
Or that x is divisible by 12.
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