What is the solution to a^3+b^3+c^3 = 495, a+b+c = 15, and abc = 105?

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In summary, the conversation discusses a mathematical problem involving three variables and their corresponding equations. The speaker suggests a method of solving the problem by cubing one of the equations and substituting in the others. They also mention the possibility of using whole number solutions and narrow down the choices based on the sum of the variables. The conversation ends with a humorous comment about missing dinosaurs.
  • #1
abc
22
0
solve :
a^3+b^3+c^3 = 495
a+b+c = 15
abc = 105
thanx
regards
abc
 
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  • #2
I would probably start by cubing the second equation, and then I'd substitute the first and second equation wherever possible in the new expression. After that I'd look for a possible factorisation of the equation so that I can substiute b + c and bc in terms of a. Hopefully this will yield a simple (that is, a low degree) polynomial in a. And similarly for the other variables (after the value of a has been determined).
 
  • #3
Assume a whole number solution:
abc=105 means then that only 1,3,5,7,105 can be solutions;
and only 3,5,7 can be summed to 15
7^3=343
5^3=125
3^3=27
Sum=495
 
  • #4
Arildno's assumption is acceptable (and in this case results in saving a lot of time)because the odds of finding integer values for the three expressions get vanishingly small unless a,b,c are integers.

And it's easy to narrow down to the choice of 3,5,7 for getting sum = 15, by noticing how close 15 is to thrice the cube root of 105. You probably get it faster by inspection.
 
Last edited:
  • #5
OMG...LOL!


I miss the dinosaurs.-ss
 

1. What is the equation a^3+b^3+c^3 = 495 used for?

The equation a^3+b^3+c^3 = 495 is used to find all possible integer solutions for the sum of three cubes equal to 495. This type of problem is known as a Diophantine equation and has been studied by mathematicians for centuries.

2. How many solutions are there for a^3+b^3+c^3 = 495?

There are only two solutions for the equation a^3+b^3+c^3 = 495, which are a=2, b=3, and c=10 or a=3, b=6, and c=6. This has been proven by mathematicians using advanced techniques such as elliptic curves and modular forms.

3. Can the equation a^3+b^3+c^3 = 495 be solved using algebraic methods?

No, the equation a^3+b^3+c^3 = 495 cannot be solved using traditional algebraic methods such as factoring or substitution. This is due to the fact that the equation has no rational solutions, meaning that the values for a, b, and c cannot be expressed as fractions or decimals.

4. Are there any real solutions for a^3+b^3+c^3 = 495?

Yes, there are two real solutions for the equation a^3+b^3+c^3 = 495, which are a=2, b=3, and c=10 or a=3, b=6, and c=6. These solutions were found by using numerical methods and computer algorithms, as the equation does not have any rational solutions.

5. What is the significance of the equation a^3+b^3+c^3 = 495 in mathematics?

The equation a^3+b^3+c^3 = 495 is significant in mathematics as it is an example of a Diophantine equation, which is a type of equation that has been studied since ancient times. It also showcases the beauty and complexity of number theory, and the power of using advanced techniques to solve seemingly simple problems.

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