Is abc/d an integer if a, b, c, and d are positive integers? (1) (a +

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

yashikaaggarwal · Answer

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)

Answer is C

rsrighosh · Answer

I am getting answer as B

The reason is:- 
as  a,b,c,d  are all positive integers and consecutive number in ascending order

then  \frac{abc}{d}  = \frac{ (a)(a+1)(a+2)}{(a+3)}  is obviously never an integer.
How C is valid

Please help  Bunuel

Bunuel · Answer

Have you tried plugging some numbers? For example, a = 3?

Archit3110 · Answer

target find whether abc/d an integer
given a,b,c,d are +ve integers
#1
(a + b + c)/d is an integer.
let a,b,c = 1 and d= 3 ; yes
and a,b,c = 1 and d = 4 ; no 
insufficient
#2
{a, b, c, d} are consecutive integers and arranged in ascending order
1,2,3,4 ; 1*2*3 /4 ; NO
3,4,5,6 ; 3*4*5 / 6; yes 
insufficient
from 1 &2
3,4,5,6 ; ; 3+4+5/6 ; yes and ; 3*4*5/6 ; 10 ;
sufficient OPTION C

GMATGuruNY · Answer

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  \frac{a+b+c}{d} =  integer, we get:
 \frac{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,  \frac{abc}{d} = \frac{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  \frac{a + b + c}{d} = \frac{2+3+4}{9} = 1  
In this case,  \frac{abc}{d} = \frac{24}{9} = \frac{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,  \frac{abc}{d} = \frac{6}{4} = \frac{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.

C

Lucky1994 · Answer

Bunuel 
Hi Bunuel,

Can you please tell me where i am going wrong by taking a=-1,b=0,c=1 and d=2?
In the above case, both statements are taken and we get the answer "No".
So won't the answer to the question is E?

Archit3110 · Answer

Lucky1994 Question says a,b,c,d are +ve integers

Posted from my mobile device

Bunuel · Answer

Notice that we are given that a, b, c, and d are  positive  integers, so none of them can be -1 or 0. But even if we were not told that the variables are positive, still a = -1, b = 0, c = 1 and d = 2, would give that abc/d = 0, which IS an integer, so we'd still have an YES answer.

chetan2u · Answer

What can you say..
1) a, b, c and d are positive integers
2) If d is prime and the greater than the other three, answer will surely be NO.
3) If d is a factor of any of the other three, answer will be YES.
4) If factors of d are also factors of product of abc, surely YES.

(1) (a + b + c)/d is an integer.
 \frac{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...... \frac{abc}{d}=\frac{1}{3} ...NO
a=b=c=2, and d=2...... \frac{abc}{d}=\frac{8}{2}=4 ...YES
Insuff

(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
Insuff

Combined..
Statement I tells us that  \frac{(a + b + c)}{d}  
=>  \frac{(b-1)+b+(b+1)}{b+2}=x......3b=(b+2)x.......(3-x)b=2x........b=\frac{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=\frac{2*2}{(3-2)}=4 , so the set = 3, 4, 5, 6...... \frac{abc}{d}=\frac{3*4*5}{6}=10 ..YES, an integer
No other possibility
Suff

C

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Is abc/d an integer if a, b, c, and d are positive integers? (1) (a +

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