gmat_winter wrote: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
Target question: Is abc/d an integer?
Statement 1: (a + b + c)/d is an integer
This statement doesn't
FEEL sufficient, so I'm going to TEST some values.
There are several values of a, b, c, and d that satisfy statement 1. Here are two:
Case a: a = 1, b = 1, c = 1 and d = 1. This meets the condition that (a + b + c)/d is an integer. In this case,
abc/d = (1)(1)(1)/1 = 1, which IS an integer
Case b: a = 1, b = 1, c = 1 and d = 3. This meets the condition that (a + b + c)/d is an integer. However, in this case,
abc/d = (1)(1)(1)/3 = 1/3, which is NOT an integer
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: https://www.gmatprepnow.com/articles/dat ... lug-values
Statement 2: {a, b, c, d} are consecutive integers and arranged in ascending order
This statement doesn't feel sufficient either, so let's TEST some values.
There are several values of a, b, c, and d that satisfy statement 2. Here are two:
Case a: a = 1, b = 2, c = 3 and d = 4. This meets the statement 2 condition. In this case,
abc/d = (1)(2)(3)/4 = 6/4 = 3/2, which is NOT an integer
Case b: a = 3, b = 4, c = 5 and d = 6. This meets the statement 2 condition. In this case,
abc/d = (3)(4)(5)/6 = 60/5 = 12, which IS an integer
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
If a, b, c, d are consecutive integers and arranged in ascending order, then we can rewrite all 4 integers in terms of a.
That is:
a = a
b = a + 1
c = a + 2
d = a + 3
Statement 1 tells us that (a + b + c)/d is an integer.
This means that [a + a+1 + a+2]/(a+3) is an integer
Simplify to see that (3a + 3)/(a + 3) is an integer
IMPORTANT: I know that (3a + 9) is divisible by (a + 3), so let's take the above expression an rewrite it to get:
(3a + 9
- 6)/(a + 3) is an integer
[notice that (3a + 9 - 6) is the same as (3a +3)]
Now rewrite the above to get: (3a + 9)/(a + 3) -
6/(a + 3) is an integer
Simplify to get: 3 -
6/(a + 3) is an integer
From this, we can conclude that
6/(a + 3) is an integer
If n is a POSITIVE integer, what can we conclude about a?
In order for
6/(a + 3) to be an integer,
a MUST EQUAL 3
If a MUST EQUAL 3, then we know that the 4 values are: a = 3, b = 4, c = 5 and d = 6
Now that we know all of the values of a, b, c and d, we can answer the
target question with certainty.
Answer =
C
Cheers,
Brent
