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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 -

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
3. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
4. EACH statement ALONE is sufficient to answer the question asked.
5. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Numbers
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.

When a positive integer 'x' is divided by a divisor 'd', the remainder is 24. What is d?
(1)    When 2x is divided by d, the remainder is 23.
(2)    When 3x is divided by d, the remainder is 22.

The correct choice is (A). Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1):  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, statement (1) tells us that the remainder is only 23. This tells us two things
1) that the divisor d is less than 48 and
2) 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 the remainder when x is divided by d is 24, the divisor cannot be 1 or 5.
So, we can conclude that 25 is the divisor.

SUFFICIENT.

Correct answer is either Choice A or Choice D.

Statement (2):  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 only 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.

nd = 50, d could either be 50 or 25 or 10 or 5 or 2.
However, as the remainder when x is divided by d is 24, d cannot be less than 24.
So, d could either be 25 or 50.

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

Not SUFFICIENT.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. Choice (A) is the correct answer.

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