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If \(x\) and \(y\) are positive integers and \(x^y = y^x + 1\), which of the following must be odd?
A. \(xy\)
B. \(x+y-1\)
C. \(x+2y\)
D. \(3x-y\)
E. \(xy+2\)
A. \(xy\)
B. \(x+y-1\)
C. \(x+2y\)
D. \(3x-y\)
E. \(xy+2\)
26 Nov 2021, 03:59
Kudos
Bookmarks
Official Solution:
If \(x\) and \(y\) are positive integers and \(x^y = y^x + 1\), which of the following must be odd?
A. \(xy\)
B. \(x+y-1\)
C. \(x+2y\)
D. \(3x-y\)
E. \(xy+2\)
CASE 1: If \(x\) is odd, then \(x^y\) is odd. Consequently, \(y^x\) becomes \(x^y -1 = odd - 1 = even\), which implies that \(y\) must be even.
CASE 2: If \(x\) is even, then \(x^y\) is even. Consequently, \(y^x\) becomes \(x^y -1 = even - 1 = odd\), which implies that \(y\) must be odd.
Now, let's evaluate the options:
A. \(xy\) is even in both cases.
B. \(x+y-1\) is even in both cases.
C. \(x+2y\) is odd in the first case and even in the second case.
D. \(3x-y\) is odd in both cases.
E. \(xy+2\) is even in both cases.
Based on the analysis of both cases, we can conclude that option D, \(3x-y\), must be odd in both cases.
Answer: D
If \(x\) and \(y\) are positive integers and \(x^y = y^x + 1\), which of the following must be odd?
A. \(xy\)
B. \(x+y-1\)
C. \(x+2y\)
D. \(3x-y\)
E. \(xy+2\)
CASE 1: If \(x\) is odd, then \(x^y\) is odd. Consequently, \(y^x\) becomes \(x^y -1 = odd - 1 = even\), which implies that \(y\) must be even.
CASE 2: If \(x\) is even, then \(x^y\) is even. Consequently, \(y^x\) becomes \(x^y -1 = even - 1 = odd\), which implies that \(y\) must be odd.
Now, let's evaluate the options:
A. \(xy\) is even in both cases.
B. \(x+y-1\) is even in both cases.
C. \(x+2y\) is odd in the first case and even in the second case.
D. \(3x-y\) is odd in both cases.
E. \(xy+2\) is even in both cases.
Based on the analysis of both cases, we can conclude that option D, \(3x-y\), must be odd in both cases.
Answer: D
