GMAT Data Sufficiency - If x is the Product of Three Consecutive Positive Integers, Which of the Following Must be True?

Question: If x is the product of three consecutive positive integers, which of the following must be true?

1. x is an integer multiple of 3
2.  x is an integer multiple of 4
3. x is an integer multiple of 6

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“If x is the product of three consecutive positive integers, which of the following must be true?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Data Sufficiency questions consist of a problem statement followed by two-three factual statements. 

Data sufficiency questions are usually refined questions for GMAT. Candidates while solving these questions do not actually understand the hints and find the questions complex. This GMAT data sufficiency question consists of five answer options.

Solution and Explanation:

There is only one approach to solve this problem.

There are certain rules in case of integers and their sums, differences and products. Given that x is the product of 3 consecutive integers, it is necessary to find which of the given statements ought to be true.

The product of k consecutive integers is divisible by k, k-1, k-2,...,2, and 1
Accordingly, it can be stated for instance, the product of any 5 consecutive integers will be divisible by 5, 4, 3, 2 and 1

Similarly, the product of any 11 consecutive integers will be divisible by 11, 10, 9, . . . 3, 2 and 1

Since x = the product of three consecutive integers, the above rule tells us that x is definitely divisible by 3 and 2.
So statement I is true

Also, if x is divisible by 3 and 2, then x is also divisible by 6.
This implies that statement III is true

However, it is important to find that x is divisible by 2.

For example it could beat the case that x = (1)(2)(3) = 6
Since 6 is not divisible by 4, it can be evaluated that statement II is not necessarily true in this case. Hence, only statements I and III is true which implies for option D as the correct answer.

Correct Answer: D

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