FACTS ESTABLISHED BY STEMS
- \(x\) is an integer
- \(y\) is an integer
- \(x > 1\)
- \(y > 1\)
QUESTION TRANSLATED
Since both \(x\) and \(y\) are integers the question "
is \(x\) a multiple of \(y\)?" really asks if
\(x = q \cdot y + r\)
where \(q\) is an integer (the quotient) and \(r\) is the remainder when you divide \(x\) by \(y\) so that \(r = 0\) which means that
\(x = q \cdot y + 0\)
which implies that
\(x = q \cdot y\)
So the question translates to:
is \(x = q \cdot y\) when \(x\), \(q\) and \(y\) are all integers, and when \(x>1\) and \(y > 1\)?STATEMENT 1
\(3y^2 + 7y = x\)
can be rewritten as
\(y(3y + 7) = x\)
however, from the question stems we know that \(y\) is an integer, which implies \(3y\) is an integer and that \(3y+7\) is also an integer:
\(y \cdot integer = x\)
which can be rewritten as
\(x = y \cdot integer\) or better yet, as
\(x = integer \cdot y\)
which if you notice is an equation of the form \(x = q \cdot y\) since \(q\) is an integer which implies that \(x\) is indeed a multiple of \(y\).
Therefore,
Statement A is sufficient.
STATEMENT 2
the statement "\(x^2 - x\)
is a multiple of \(y\)" must be rewritten as "\(x (x - 1)\)
is a multiple of \(y\)" which implies that
either \(x\) is a multiple of \(y\) or \(x-1\) is a multiple of \(y\)
the sub-statement \(x\) is a multiple of \(y\) implies that \(x = q \cdot y\) which would answer the question, but we still need to figure out if the second sub-statement answers the question with the same answer as the first sub-statement.
the second sub-statement \(x-1\) is a multiple of \(y\) implies that
\(x-1 = q \cdot y\)
which can be rewritten as
\(x = q \cdot y + 1\)
which is an equation of the form \(x = q \cdot y + r\) when \(r=1\) which implies that when \(x\) is divided by \(y\) we get a remainder of 1, meaning that \(x\) is NOT a multiple of \(y\).
Since both sub-statements are contradictory, this means that when \(x^2 - x\) is a multiple of \(y\), \(x\) is not always a multiple of \(y\). meaning that we cannot determine whether \(x\) is
always a multiple of \(y\) according to Statement 2 alone.
Therefore,
Statement 2 is not sufficient.
ANSWER
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.