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checkitagain
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Determine all solutions for [tex]\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \ = \ 5, [/tex] [tex]where \ \ a, \ b, \ and \ \ c \ \ are \ \ positive \ \ integers, \ \ and \ \ a <b < c.[/tex]
Wilmer said:1,2,4
2,4,8
3,6,12
...
n,2n,4n
...CaptainBlack said:> It is quite obvious that any multiple of a solution will also be a solution,
> so in essence you have a single solution here.
Thank you, Sir.
> Are there any solutions that are not a multiple of 1,2,4?
None with c < 1000
Wilmer said:...
The equation is a^2 + b^2 + c^2 = abc.
This means that the values of a, b, and c must be whole numbers that are greater than 0.
Yes, all three values must be positive integers and the equation must hold true for all three values.
It is not possible to determine the exact number of solutions without knowing the specific values of a, b, and c. However, there are an infinite number of possible solutions.
Equations like this one can have real-world applications in fields such as number theory and cryptography. Finding solutions can also help us better understand and explore the properties of numbers.