Is abc/d an Integer if a, b, c, and d are Positive Integers? GMAT Data Sufficiency

Question: Is abc/d an integer if a, b, c, and d are positive integers?

(1) (a + b + c)/d is an integer.
(2) {a, b, c, d} are consecutive integers and arranged in ascending order

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.

“Is abc/d an integer if a, b, c, and d are positive integers?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide". 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.

Solution and Explanation:
Approach Solution 1:
Given : a, b, c, and d are positive integers?
To find : Is abc/d an integer?

(1) (a + b + c)/d is an integer.
Case 1: Let A = 2, B = 4, C = 6 and D = 3
A+B+C/D = 12/3 =4 and ABC/D = 48/3 = 16 (integer)

Case 2: Let A = 2, B = 4, C = 9 and D = 5
A+B+C/D = 15/5 =3 and ABC/D = 72/5 = 14.2 (Non-integer)
Insufficient

(2) {a, b, c, d} are consecutive integers and arranged in ascending order
there are many possible values for A,B,C and D
Insufficient

Statement 1 and 2:
There is only one possible combination satisfying both constraint, when consecutive 4 no. set is 3,4,5,6 => A+B+C/D is an integer => 12/6 = 2
also ABC/D = 60/6 = is an integer.
SUFFICIENT

Correct Answer: C

Approach Solution 2:
Since each statement seems likely to be insufficient on its own, let's test when the statements are combined.

Case 1, when the statements are combined:
Since a, b, c and d are consecutive integers, a+b+c = 3b and d=b+2
Since a+b+c/d = integer, we get:
3b/b+2 = integer

Thus, 3b must be a multiple of b+2, yielding the following options:
3b= b+2
3b=2b+4
3b=3b+6
3b=4b+8
And so on.

Only the option in green yields positive values for a, b, c and d:
3b=2b+4
b=4, with the result that a=3, b=4, c=5 and d=6
In this case, abc/d = 60/6 =10, so the answer to the question stem is YES.

Statement 1:
In Case 1, the answer to the question stem is YES.

Case 2: a=2, b=3, c=4, and d=9, with the the result that a+b+c/d = 2+3+4/9 =1

In this case, abc/d = 24/9 = 8/3, so the answer to the question stem is NO.

Since the answer is YES in Case 1 but NO in Case 2, INSUFFICIENT.

Statement 2:
In Case 1, the answer to the question stem is YES.

Case 3: a=1, b=2, c=3, and d=4
In this case, abc/d = 6/4 = 3/2, so the answer to the question stem is NO.

Since the answer is YES in Case 1 but NO in Case 3, INSUFFICIENT.

Statements combined:
Only Case 1 satisfies both statements.
In Case 1, the answer to the question stem is YES.
SUFFICIENT.

Correct Answer: C

Approach Solution 3:
(1) (a + b + c)/d is an integer.
a+b+c/d=y, where y is an integer.

But d could be a prime greater than the other 3.....a=b=c=1, and d=3......abc/d = 13...NO
a=b=c= 2, and d= 2......abc/d= 82= 4...YES
Insufficient

(2) {a, b, c, d} are consecutive integers and arranged in ascending order
So numbers are b-1, b, b+1, b+2…
If numbers are 1, 2, 3, 4...NO
If numbers are 3, 4, 5, 6...YES
Insufficient

Combined..
Statement I tells us that (a+b+c)/d
=> (b−1)+b+(b+1)/b+2= x......3b= (b+2)x.......(3−x)b= 2x........b= 2x/3−x
Since b is positive, x can be
a)x=1...b=1 so the set = 0, 1, 2, 3, but a cannot be 0

b) x= 2..b= 2∗2/(3−2)=4, so the set = 3, 4, 5, 6......abc/d= 3∗4∗56=10..YES, an integer
No other possibility
Sufficient

Correct Answer: C

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