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.

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.

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 11: 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

**What kind of an answer will the question fetch?**

The question is an **"Is"** question. Answer to an "is" question 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.

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?**

A number is divisible by 21 if it is divisible by 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. (14k is divisible by 7. When 4 is divided by 7, the remainder is 4.)

This number is definitely not divisible by 7.

To be divisible by 21, the number must be divisible by both 3 and 7. This number is 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.

To be divisible by 21, the number must be divisible by both 3 and 7. This number 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 also sufficient.**

Eliminate choice A.

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1. Number Systems | Types of Numbers | Chart

2. Number Properties | Rational & Irrational Numbers

3. GMAT Number Properties | Indices & Rule of Exponents

4. Number Systems | Surds & Conjugates

5. Number Properties | Tests of divisibility

6. Number Properties | How to check whether a number is prime?

7. Number Properties | How to prime factorize a number?

8. GMAT Number Theory | Prime factorization | Properties of squares & cubes

9. Number Properties | What is HCF? | How to find HCF?

10. Number Properties | What is LCM? | How to find LCM?

11. 3 important properties of LCM & HCF | LCM & HCF of fractions

12. Number Properties | When to use LCM and HCF?

13. Number Theory | How to find number of factors?

14. Number Theory | Number of ways to express as a product of 2 factors

15. Number Theory | Sum of all factors of a number

16. Number Theory | Product of all factors of a number

17. Number Theory | Remainders of sum & product

18. Number Theory | Remainder of dividing x^{n} by 'd'

19. Polynomials | Remainder when a monomial divides it

20. Number Theory | Highest power of a prime that divides factorial of 'n'

21. Number Theory | Highest power of a composite number that divides factorial of 'n'

22. Number Theory | Number of trailing zeroes in a number

23. Number Theory | Unit digit of higher powers of numbers

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