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anemone
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Find all integer solutions $(a,\,b)$ satisfying $a^4+79+b^4=48ab$.
anemone said:Find all integer solutions $(a,\,b)$ satisfying $a^4+79+b^4=48ab$.
kaliprasad said:we make following observations
No 1) both are positive / -ve
No 2) one is odd and another even as RHS is even
No 3) The lower number is $< 5$ as $2 * 5^4 >= 1200$
No 4) x mod 3 cannot be zero nor y mod 3 as 79 mod 3 = 1 and RHS mod 3 = 0
No 5) if $(x,y) $ is a solution then $(-x,-y)$ is a solution
based on this we need to check (1,2), (1,4) so on
(2,5) so on
(4,5) so on
using above 3 conditions we get (1,2) a solution
(1,4) gives LHS = 336 > RHS
(2,5) gives LHS larger
(4,5) give LHS = RHS
so solution set = $(1,2).(2,1),(4,5),(5,4),(-1,-2),(-2,-1),(-4,-5),(-5,-4)$
anemone said:Well done, kaliprasad! And thanks for participating!:)
kaliprasad said:Thanks for the same. I would like to see your solution Anemone
To find all integer solutions to a given problem, one must first identify the variables and constraints in the problem. Then, a systematic approach such as substitution or elimination can be used to solve for the values of the variables that satisfy all of the constraints. This process may require multiple steps and may involve testing different values until all solutions have been found.
Yes, it is possible to have an infinite number of integer solutions to a given problem. This can occur when the constraints allow for a range of values for the variables, such as in the case of an inequality, or when there are multiple variables with no restrictions on their values.
Yes, it is possible for a problem to have no integer solutions. This can happen when the constraints are too restrictive and do not allow for any values of the variables to satisfy all of the conditions.
Yes, there are some strategies that can make the process of finding all integer solutions more efficient. For example, using a systematic approach like substitution or elimination can help to eliminate trial and error. Additionally, breaking the problem down into smaller parts or using patterns and relationships between variables can also make the process more efficient.
To check your answers and make sure you have found all integer solutions, you can substitute the values you have found into the original problem and see if they satisfy all of the constraints. You can also use a graphing or algebraic tool to visualize and verify your solutions.