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If x, y, z are distinct positive integers such that x2 + y2 = z2( 1 + xy) then how many ordered pairs of x, y and z satisfy this equation?
Correct answer is '0'. Can you explain this answer?
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Verified Answer
If x, y, z are distinct positive integers such that x2 + y2 = z2( 1 + ...
x2 + y2 > (1 + xy) for distinct x and y.
Only for z2 > 1, x2 + y2 = z2(1 + xy) The equation is symmetric in x and y, so we can assume (without loss of generality) x = y
Only for z2 > 1, x2 + y2 = z2(1 + xy) The equation is symmetric in x and y, so we can assume (without loss of generality) x = y
2x2 = z2(1 + x2) ... (i)
As x is positive integer, we can write x2 < x2 + 1
x2z2 < z2(x2 + 1) ... (ii)
Using (ii) in (i) 2x2 > x2z2
z2 < 2
z = 1 (as z is positive integer)
x2 + y2 = (1 + xy) This is true only for x = y = 1 Thus, there is no pair of distinct positive integers that satisfy the given equation.
Answer: 0
Answer: 0
Most Upvoted Answer
If x, y, z are distinct positive integers such that x2 + y2 = z2( 1 + ...
Given equation: x^2 - y^2 = z^2(1 - xy)
To find the number of ordered pairs (x, y, z) that satisfy this equation, let's analyze the equation step by step.
Step 1: Simplify the equation
Using the identity a^2 - b^2 = (a + b)(a - b), we can simplify the equation as follows:
(x + y)(x - y) = z^2(1 - xy)
Step 2: Analyze the factors
The given equation has two factors on the left side: (x + y) and (x - y).
Since x, y, and z are distinct positive integers, let's consider the possible values for these factors.
Case 1: x + y = 1 and x - y = z^2(1 - xy)
If x + y = 1, it means that x = 1 and y = 0 or x = 0 and y = 1. However, the question states that x, y, and z are distinct positive integers, so this case is not valid.
Case 2: x + y = z^2 and x - y = 1 - xy
If x + y = z^2, it implies that z^2 is the sum of two positive integers. However, the sum of two positive integers cannot be a perfect square unless both integers are zero. Since x, y, and z are distinct positive integers, this case is not valid.
Step 3: Conclusion
From the analysis above, we can see that there are no possible values for (x, y, z) that satisfy the given equation.
Therefore, the correct answer is '0' - there are no ordered pairs (x, y, z) that satisfy the equation x^2 - y^2 = z^2(1 - xy).
To find the number of ordered pairs (x, y, z) that satisfy this equation, let's analyze the equation step by step.
Step 1: Simplify the equation
Using the identity a^2 - b^2 = (a + b)(a - b), we can simplify the equation as follows:
(x + y)(x - y) = z^2(1 - xy)
Step 2: Analyze the factors
The given equation has two factors on the left side: (x + y) and (x - y).
Since x, y, and z are distinct positive integers, let's consider the possible values for these factors.
Case 1: x + y = 1 and x - y = z^2(1 - xy)
If x + y = 1, it means that x = 1 and y = 0 or x = 0 and y = 1. However, the question states that x, y, and z are distinct positive integers, so this case is not valid.
Case 2: x + y = z^2 and x - y = 1 - xy
If x + y = z^2, it implies that z^2 is the sum of two positive integers. However, the sum of two positive integers cannot be a perfect square unless both integers are zero. Since x, y, and z are distinct positive integers, this case is not valid.
Step 3: Conclusion
From the analysis above, we can see that there are no possible values for (x, y, z) that satisfy the given equation.
Therefore, the correct answer is '0' - there are no ordered pairs (x, y, z) that satisfy the equation x^2 - y^2 = z^2(1 - xy).
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Community Answer
If x, y, z are distinct positive integers such that x2 + y2 = z2( 1 + ...
x2 + y2 > (1 + xy) for distinct x and y.
Only for z2 > 1, x2 + y2 = z2(1 + xy) The equation is symmetric in x and y, so we can assume (without loss of generality) x = y
Only for z2 > 1, x2 + y2 = z2(1 + xy) The equation is symmetric in x and y, so we can assume (without loss of generality) x = y
2x2 = z2(1 + x2) ... (i)
As x is positive integer, we can write x2 < x2 + 1
x2z2 < z2(x2 + 1) ... (ii)
Using (ii) in (i) 2x2 > x2z2
z2 < 2
z = 1 (as z is positive integer)
x2 + y2 = (1 + xy) This is true only for x = y = 1 Thus, there is no pair of distinct positive integers that satisfy the given equation.
Answer: 0
Answer: 0
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