The given question is a challenging GMAT data sufficiency question testing concepts in Statistics. Concepts include Progressions and Range.

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 28: What is the range of 3 positive integers a, b, and c ?

**Statement 1**: 21a = 9b = 7c

**Statement 2**: a + 8, b, and c, in that order are in AP.

INR

**Statement 1**: 21a = 9b = 7c

Let 21a = 9b = 7c = n

Because a, b, and c are positive integers, 'n' is a common multiple of 21, 9, and 7.

The least value for 'n' is the LCM of 21, 9, and 7 = 63.

So, 'n' is a multiple of 63 or n = 63k.

So, a = 3k, b = 7k and c = 9k. The range of the numbers is 9k – 3k = 6k

Without knowing the value of k we **cannot find the range of the 3 numbers**.

Hence, statement 1 is not sufficient.

__Eliminate answer option A and D__.

**Statement 2**: a + 8, b, and c, in that order are in AP.

__Approach __: Counter example

__Example __: a = 1, b = 10, and c = 11. a + 8 = 9

So, 9, 10, 11 are in AP. Range = 11 - 1 = 10

__Counter Example __: a = 1, b = 11, c = 13. a + 8 = 9

So, 9, 11, 13 are in AP. Range = 13 - 1 = 12

Counter example exists.

We are **not able to find a UNIQUE value** for the range using Statement 2.

Hence, statement 2 is not sufficient.

__Eliminate answer option B__.

**Statements**: **Statement 1**: 21a = 9b = 7c

**Statement 2**: a + 8, b, and c in that order are in AP.

**Key inferences**:

__From statement 1__: we know a = 3k, b = 7k and c = 9k

__From statement 2__: we know c - b = b - (a + 8) (because a + 8, b, and c are in AP) ... (1)

Substiute c = 9k, b = 7k and a = 3k in equation (1)

So, 9k - 7k = 7k - (3k + 8)

2k = 4k - 8

or k = 4

Therefore, the range (9k - 3k) = 6k = 6(4) = 24

We are **able to find a UNIQUE value** for range using the statements together.

Statements together are sufficient.

__Eliminate answer option E__.

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