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26 May 2017, 04:47
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\(x, y,\) and \(n\) are positive integers. If \(x\) and \(y\) are factors of \(n\), which of the following must be an integer?
I. \(\frac{n}{x+y}\) II. \(\frac{n}{xy}\) III. \(\frac{n}{x} + \frac{n}{y}\)
A. I only
B. II only
C. III only
D. I and II only
E. none of them
I. \(\frac{n}{x+y}\) II. \(\frac{n}{xy}\) III. \(\frac{n}{x} + \frac{n}{y}\)
A. I only
B. II only
C. III only
D. I and II only
E. none of them
26 May 2017, 04:48
Kudos
Bookmarks
Official Solution:
\(x, y,\) and \(n\) are positive integers. If \(x\) and \(y\) are factors of \(n\), which of the following must be an integer?
I. \(\frac{n}{x+y}\) II. \(\frac{n}{xy}\) III. \(\frac{n}{x} + \frac{n}{y}\)
A. I only
B. II only
C. III only
D. I and II only
E. none of them
\(x\) and \(y\) are factors of \(n\), so \(x = 4\), \(y = 6\) and \(n = 12\) is possible. If so,
I. \(\frac{n}{x+y} = \frac{12}{4+6} = \frac{12}{10} = \frac{6}{5}\) (X)
II. \(\frac{n}{xy} = \frac{12}{(4)(6)} = \frac{12}{24} = \frac{1}{2}\) (X)
III. Since \(x\) and \(y\) are factors of \(n, n = xa = yb\) (\(a, b\) are any positive integers) If so, \(\frac{n}{x} + \frac{n}{y} = a+b = integer (O)\).
Therefore, C is the answer
Answer: C
\(x, y,\) and \(n\) are positive integers. If \(x\) and \(y\) are factors of \(n\), which of the following must be an integer?
I. \(\frac{n}{x+y}\) II. \(\frac{n}{xy}\) III. \(\frac{n}{x} + \frac{n}{y}\)
A. I only
B. II only
C. III only
D. I and II only
E. none of them
\(x\) and \(y\) are factors of \(n\), so \(x = 4\), \(y = 6\) and \(n = 12\) is possible. If so,
I. \(\frac{n}{x+y} = \frac{12}{4+6} = \frac{12}{10} = \frac{6}{5}\) (X)
II. \(\frac{n}{xy} = \frac{12}{(4)(6)} = \frac{12}{24} = \frac{1}{2}\) (X)
III. Since \(x\) and \(y\) are factors of \(n, n = xa = yb\) (\(a, b\) are any positive integers) If so, \(\frac{n}{x} + \frac{n}{y} = a+b = integer (O)\).
Therefore, C is the answer
Answer: C
Moderator:
