AGMAT DS question in Number Properties and number theory. Concept covered is remainders and divisors.

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 14: When a positive integer 'x' is divided by a divisor 'd', the remainder is 24. What is d?

- When 2x is divided by d, the remainder is 23.
- When 3x is divided by d, the remainder is 22.

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

The question is a "What is the value?" question. For questions asking for a value, the answer should be a number.

**When is the data sufficient?**

The data is sufficient if we are able to get a UNIQUE answer for the value of 'd' from the information in the statements.

If either the statements do not have adequate data to determine the value of 'd' or if more than one value of 'd' exists based on the information in the statement, the data is NOT sufficient.

**What do we know from the question stem?**

'x' is a positive integer. Dividing x by d leaves a remainder of 24.

So, the value of 'd' is more than 24.

Evaluate Statement (1) ALONE: When 2x is divided by d, the remainder is 23.

The question stem states that when x is divided by d, the remainder is 24.

Therefore, when 2x is divided by d, the remainder should be 2 * 24 = 48.

However, from statement (1) we know that the remainder is 23. We can infer the following from the question stem and statement 1:

- the divisor d is less than 48
- the divisor is at least 25 and
- 48 divided by divisor d should leave a remainder of 23.

i.e., 48 = nd + 23 or nd = 25.

The possible values for d are 1, 5 and 25.

However, as d is at least 25, the divisor cannot be 1 or 5.

So, we can conclude that 25 is the divisor.

**Statement 1 ALONE is sufficient.**

Eliminate choices B, C and E. Choices narrow down to A or D.

Evaluate Statement (2) ALONE: When 3x is divided by d, the remainder is 22.

If x leaves a remainder of 24 when divided by d, then 3x will leave a remainder of 3 * 24 = 72 when divided by d.

However, the remainder is 22.

This tells us that the divisor is less than 72 and that 72 divided by d leaves a remainder of 22.

So, 72 = n * d + 22

Or nd = 72 - 22 = 50

If nd = 50, d could be 50 or 25 or 10 or 5 or 2.

However, from the question stem we have deduced that the divisor is at least 25. So, d cannot be 10, 5 and 2.

But, d could be 25 or 50.

From statement 2, we are unable to deduce a unique value for d.

**Statement 2 ALONE is NOT sufficient.**

Eliminate choice D.

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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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