The given question is a GMAT Hard Math data sufficiency question. This question is a GMAT arithmetic practice question that tests your understanding of properties of positive integers from the topic Number Properties.

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 36: What is the value of (m + n) if m and n are positive integers?

**Statement 1**: m^{2} – n^{2} = 105

**Statement 2**: Neither m nor n is divisible by 8.

INR

**Statement 1**: m^{2} – n^{2} = 105

m and n are positive integers.

m^{2} – n^{2} can be written as (m – n)(m + n). Therefore, (m – n)(m + n) = 105

Both (m – n) and (m + n) have to be positive integers. (Why? Because (m + n) is positive and the product of (m + n) and (m - n) is positive.)

The ways of expressing 105 as a product of 2 positive integers is listed in the table given below.

(m - n) | (m + n) |
---|---|

1 | 105 |

3 | 35 |

5 | 21 |

7 | 15 |

So, (m + n) could be 105, 35, 21, or 15.

We are **not able to answer the question with a UNIQUE value**.

Hence, statement 1 alone is not sufficient.

__Eliminate answer options A and D__.

**Statement 2**: Neither m nor n is divisible by 8.

Infinite possibilities exist for this condition.

We are **not able to answer the question with a UNIQUE value**.

Hence, statement 2 alone is not sufficient.

__Eliminate answer option B__.

**Statement 1**: m^{2} – n^{2} = 105

**Statement 2**: Neither m nor n is divisible by 8.

**From statement 1**: (m – n) (m + n) = 105 and both (m – n) and (m + n) are positive integers.

(m - n) | (m + n) | m | n |
---|---|---|---|

1 | 105 | 53 | 52 |

3 | 35 | 19 | 16 |

5 | 21 | 13 | 8 |

7 | 15 | 11 | 4 |

**From Statement 2**: We can eliminate (19, 16) and (13, 8).

We still have two values for (m, n): (53, 52) and (11, 4)

Therefore, (m + n) could be 105 or 15.

Despite combining both the statements, we are **not able to find a UNIQUE value for (m + n)**.

Statements TOGETHER are NOT sufficient.

__Eliminate answer option C__.

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