Are the positive integers x and y consecutive?

You can simplify the first statement in following way:
(1) \(x^2-Y^2=2y+1\)
i.e. \(x^2 = Y^2 + 2y+1\)
i.e. \(x^2 = (y+1)^2\)
i.e. \(x=|y+1| = y + 1\)(taking positive root as both x & y are positive integers)
Hence (1) is SUFFICIENT as x & y are consecutive

(2) \(x^2-xy-x=0\)
i.e. \(x(x-y-1)=0\)
i.e. \(x=0\) or \(x=y+1\)
As x is positive integer x<>0, thus \(x=y+1\)
Hence (2) is SUFFICIENT.

Choice (D) is the answer.

The question is basically asking whether \(x=y+1\)
Statement 1)
\(x^2 - y^2=2y+1\) can be written as \((x+y)(x-y)=2y+1\).----equation 1
If we put \(x=y+1\), then LHS must be equal to RHS.

Equation 1 can be written, after substituting x=y+1, as \((2y+1)(1)=2y+1\). They are equal. Hence x and y are consecutive.

Statement 2)
\(x(x-y-1)=0\)
The above equation can be equal to 0 only when \(x-y=1\) because x is given to be positive.

Hope I am correct.
+1D

Similar question to practice: if-x-y-and-z-are-positive-integers-where-x-y-and-z-x-126859.html
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Statement 1) \(x^2 - y^2 = 2y + 1\)

\(x^2 = y^2+2y+1\)
\(x^2 = (y+1)^2\)
Since x and y are positive x = y+1
x-y = 1

When the difference between two numbers is 1 then they both must be consecutive integers. So statement 1) SUFFICIENT.

Statement 2) \(x^2 - xy - x = 0\)
\(x (x-y-1) = 0\)

Either x = 0 or (x-y-1) = 0
Since it is already mentioned that x is a positive integer, here x cannot be equal to zero.
So x-y-1 =0
x-y = 1

When the difference between two numbers is 1 then they both must be consecutive integers. So statement 2) SUFFICIENT.

OPTION: D

From 1:
put x=2 and y=1, expression is satisfied, sufficient

From 2:
again check with x=2 and y=1 , sufficient

IMO Clearly D

why is the assumption that x=y+1 and not y = x+1 ?

You are correct that x and y are consecutive if y = x+1, as well as if x = y+1. So, if we can use a statement to prove that either y = x+1 or x = y+1, then that statement is sufficient. In this question, each statement can be used to prove that x = y+1, so each statement is sufficient. However, if you could use each statement to prove that y = x+1, then each statement would be sufficient in that case as well.

Please let me know if you have any more questions!

I wanted to point out that, in your solution, you have just found one example of consecutive integers that is consistent with each statement. This means that x and y could be consecutive integers, but is not enough to show that x and y must be consecutive integers, so at this point we would not be sure that each statement is sufficient. Doing the algebra, as shown in other solutions, allows us to see that each statement tells us that x = y+1, which means that x and y must be consecutive integers, so each statement is sufficient.

I'm guessing that you may have just been moving through this question too quickly, but I thought I'd use this opportunity to point out a couple of common issues on Data Sufficiency that could lead to approaching a question in the way shown in this solution:

1) It's important to remember to assume that each statement is true, and see if you can get to just one answer to the question based on that statement. Especially in Yes/No questions like this, it can be tempting to start by assuming a "Yes" answer to the question (in this question, assuming that x and y are consecutive integers), and then seeing if this is consistent with a statement. This just tells us that the answer to the question could be "Yes". However, to prove that it must be "Yes", we need to start with each statement, and see if we can show that the answer to the question must be "Yes" if that statement is true.

2) When using number picking on Data Sufficiency, it's not enough to pick a single set of numbers that is consistent with a given statement. Once we have picked one set of numbers that is consistent with a statement and gotten one answer to the question, our goal is to keep picking numbers until one of the following happens:
(a) We pick another set of numbers that is consistent with the statement but gives another answer to the question (this means that the statement is not sufficient).
(b) We pick enough numbers that we can see conceptually that the statement always leads to the same answer to the question (this means that the statement is sufficient).
(c) We have tried all possible numbers that are consistent with the statement, and we always get the same answer to the question (this means that the statement is sufficient). This is possible for some statements, or combinations of statements, that limit us to a small number of possibilities, but this is not possible for either of the statements for this question, because there are an infinite number of values of x and y that are consistent with each statement.

I hope that the above is helpful. Please let me know if you have any questions!

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