If x and y are integers great than 1, is x a multiple of y?

Please find the solution as mentioned in attachment

Target question: Is x a multiple of y?
Asking whether x is a multiple of y is the same as asking whether x = (y)(some integer)
For example, 12 is a multiple of 3 because 12 = (3)(4)
So, let's rephrase the question as...
REPHRASED target question: Does x = (y)(some integer)?

Statement 1: 3y² + 7y = x
Factor to get x = y(3y + 7)
If y is an integer, then (3y + 7) must be an integer
In other words: x = y(some integer)
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: x² - x is a multiple of y
There are several values of x and y that satisfy this condition. Here are two:
Case a: x = 4 and y = 2 (this satisfies statement 2 because x² - x = 12, and 12 is a multiple of 2). In this case, x IS a multiple of y
Case b: x = 5 and y = 2 (this satisfies statement 2 because x² - x = 20, and 20 is a multiple of 2). In this case, x is NOT a multiple of y
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

If 20 is a multiple of 2 then how X is not a multiple of Y?

When x = 5 and y = 2, we see that x² - x = 20, and 20 is a multiple of 2
In other words, x² - x is a multiple of 2
However, the target question asks b]Is x a multiple of y?[/b](not Is x² - x a multiple of y?)

If x = 5 and y = 2, then x is NOT a multiple of y

Cheers,
Brent

Would this be a valid way of looking at it?

Target Question: is x = y*I (where I is an integer)

Statement I: same as all of your solutions above - SUFF.

Statement II :

x^2 - x = y*I (where I is some integer)
x (x - 1) = y*I
x = (y*I)/(x-1)
x = y* (I/(x-1))

Now, we know (x-1) is definitely an integer and so is I, but we have no idea whether the ratio of I and (x-1) is ALSO an integer. Thus, statement II is INSUFF.

Thoughts?

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