The Product of Three Distinct Positive Integers is Equal to the Square GMAT Data Sufficiency

Question: The product of three distinct positive integers is equal to the square of the largest of the three numbers, what is the product of the two smaller numbers?

1. The average (arithmetic mean) of the three numbers is \(\frac{3}{4}\).
2. The largest number of the three distinct numbers is 24.

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.

Solution and Explanation:

Approach Solution (1):

Let the three integers be a, b, and c, where 0 < a < b < c
Given: \(abc=c^2\) so, ab = c
Question: ab = c = ?

(1) The average of the three numbers is \(\frac{3}{4}\)

This basically means that a + b + c = 34. Substitute the value of c with ab to get a + b + ab = 34
From that we can write (a + 1)(b + 1) = 35
Now, since a and b are integers, then a + 1 = 5 and b + 1 = 7
a = 4 and b = 6, hence ab = 24
Sufficient

(2) The largest number of the three distinct numbers is 24. Directly gives the value of c
Sufficient

Correct Option: D

Approach Solution (2):

The question stem says three distinct positive integers and the product is equal to the square of the highest number which means the product is equal to the highest number
Let’s say abc, c being the highest number
\(abc=c^2\)

Which implies that ab = c
First choice explanation:
a + b + c = 34
adding 1 on both sides, we get:
a + b + c + 1 = 34 + 1
we know ab = c
a + b + ab + 1 = 35
a + ab + b = 34
a(1 + b) + (1 + b) = 35
Taking (1 + b) out
(1 + b) (1 + a) = 35
Factors of 35 = 1, 5, 7, 35
(1, 35) cannot be the pair as a or b is a positive integer
(5, 7) is the pair
Which means (4, 6) is a pair for (a, b)
Hence the product is 24

Correct Option: D

Approach Solution (3):

As we know that:
a + b + ab = 34
Add +1 to both the sides:
a + b + ab + 1 = 34 + 1
a (b + 1) + (b + 1) = 35
(b + 1) (a + 1) = 35
S2, the largest number of the three distinct numbers is 24. Directly gives the value of c
Sufficient

Correct Option: D

“The product of three distinct positive integers is equal to the square of the largest of the three numbers, what is the product of the two smaller numbers?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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