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26 Apr 2024, 08:10
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If \(x\) and \(y\) are integers and \(((x - 1)(y - 1))! = 2\), which of the following could be a value of \(xy\)?
I. -6
II. 0
III. 6
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
I. -6
II. 0
III. 6
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
26 Apr 2024, 08:10
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Bookmarks
Official Solution:
If \(x\) and \(y\) are integers and \(((x - 1)(y - 1))! = 2\), which of the following could be a value of \(xy\)?
I. -6
II. 0
III. 6
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
Only \(2! = 2\), so we have that \((x - 1)(y - 1) = 2\).
Since \(x\) and \(y\) are integers, we can have the following four cases:
\((x - 1)=2\) and \((y - 1) = 1\). This gives \(x=3\) and \(y=2\).
\((x - 1)=1\) and \((y - 1) = 2\). This gives \(x=2\) and \(y=3\).
\((x - 1)=-2\) and \((y - 1) = -1\). This gives \(x=-1\) and \(y=0\).
\((x - 1)=-1\) and \((y - 1) = -2\). This gives \(x=0\) and \(y=-1\).
The first two cases yield \(xy=6\), and the last two cases yield \(xy=0\).
Answer: D
If \(x\) and \(y\) are integers and \(((x - 1)(y - 1))! = 2\), which of the following could be a value of \(xy\)?
I. -6
II. 0
III. 6
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
Only \(2! = 2\), so we have that \((x - 1)(y - 1) = 2\).
Since \(x\) and \(y\) are integers, we can have the following four cases:
\((x - 1)=2\) and \((y - 1) = 1\). This gives \(x=3\) and \(y=2\).
\((x - 1)=1\) and \((y - 1) = 2\). This gives \(x=2\) and \(y=3\).
\((x - 1)=-2\) and \((y - 1) = -1\). This gives \(x=-1\) and \(y=0\).
\((x - 1)=-1\) and \((y - 1) = -2\). This gives \(x=0\) and \(y=-1\).
The first two cases yield \(xy=6\), and the last two cases yield \(xy=0\).
Answer: D
