The weekend is a great time to get some GMAT studying done. To help you, today we share a sample problem and an explanation of its solution provided by Manhattan GMAT:
Problem
If a, b, and c are positive integers, with a < b < c, are a, b, and c consecutive integers?
(1) 1/a – 1/b = 1/c
(2) a + c = b2 – 1
A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D: EACH statement ALONE is sufficient.
E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.
Solution
The question can be rephrased “Is b = a + 1 and is c = a + 2?”
One way to approach the statements is to substitute these expressions involving a and solve for a. Since this could involve a lot of algebra at the start, we can just substitute a + 1 for b and test whether c = a + 2, given that both are integers.
Statement 1: SUFFICIENT.
Following the latter method, we have
1/a – 1/(a + 1) = 1/c
(a + 1)/[a(a + 1)] – a/[a(a + 1)] = 1/c
1//[a(a + 1)] = 1/c
a2 + a = c
Now we substitute a + 2 for c and examine the results:
a2 + a = a + 2
a2 = 2
a is the square root of 2. However, since a is supposed to be an integer, we know that our assumptions were false, and a, b, and c cannot be consecutive integers.
We can now answer the question with a definitive “No,” making this statement sufficient.
We could also test numbers. Making a and b consecutive positive integers, we can solve the original equation (1/a – 1/b = 1/c). The first 4 possibilities are as follows:
1/1 – 1/2 = 1/2
1/2 – 1/3 = 1/6
1/3 – 1/4 = 1/12
1/4 – 1/5 = 1/20
Examining the denominators, we can see that c = ab. None of these triples so far are consecutive, and as a and b get larger, c will become more and more distant, leading us to conclude that a, b, and c are not consecutive.
Statement 2: SUFFICIENT
Let’s try substituting (a + 1) for b and (a + 2) for c.
a + a + 2 = (a + 1)2 – 1
2a + 2 = a2 + 2a
2 = a2
Again, we get that a must be the square root of 2. However, we know that a is an integer, so the assumptions must be false. We can answer the question with a definitive “No,” and so the statement is sufficient.
The answer is D: Each statement is sufficient.
