GMAT Data Sufficiency  Divisibility
Question 11
2. Test of divisibility. 2 concepts
The GMAT DS question given below is Number Properties question and the concept covered is test of divisibility of numbers and remainders of the division.
Directions
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  Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
 Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
 BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
 EACH statement ALONE is sufficient to answer the question asked.
 Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
All numbers used are real numbers.
Figures
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.
Note
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
Is the positive integer X divisible by 21?
 When X is divided by 14, the remainder is 4
 When X is divided by 15, the remainder is 5
Explanatory Answer

What kind of an answer will the question fetch?
The question is an "Is" question. For "Is" questions, the answer will be YES or NO.
When is the data sufficient?
The data is sufficient if we get a DEFINITE YES or a DEFINITE NO as an answer from the information in the statements.
If we get a MAYBE as an answer, the data is NOT sufficient.
Do we have any more information about 'X' from the question stem?
The question stem states that 'X' is a positive integer
What kind of numbers will be divisible by 21?
Any number that is divisible by 21 will be divisible by both 3 and 7.

Statement 1: When X is divided by 14, the remainder is 4
The number is, therefore, of the form 14k + 4.
It will leave a remainder of 4 when divided by 7.This number is definitely not divisible by 7.
Hence, X is not divisible by 21.
We have a DEFINITE NO as the answer to the question using statement 1.
Statement 1 ALONE is sufficient.
Eliminate choices B, C and E. Choices narrow down to A or D.

Statement 2: When X is divided by 15, the remainder is 5
The number X is of the form 15m + 5
Therefore, the number will leave a remainder of 2 when divided by 3.
Hence, it is not divisible by 3.
Hence, X is not divisible by 21.
We have a DEFINITE NO as the answer to the question using statement 2 as well.
Statement 2 ALONE is sufficient.
Each statement is INDEPENDENTLY sufficient. Eliminate choice A. Choice D is the answer.
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