Question

Find the natural number with the property that it can be expressed as the sum of the cubes of two natural numbers in two different ways?

Answer 1

Hint: In order to answer this question, to know the natural numbers with the property that they can be expressed as the sum of the cubes of two natural numbers in two different ways, we will go through the trial and error method to solve this problem.

Complete step by step solution:
There is only one way to follow according to the question or to writing them-
The number that can be expressed in sum of two cubes in 2 different ways is given by trial and error method:-
 $ {1^3} + {2^3} = 9 $
 $ {1^3} + {3^3} = 28 $
 $ {1^3} + {4^3} = 65 $
 $ {1^3} + {5^3} = 126 $
 $ {1^3} + {6^3} = 217 $
 $ {1^3} + {7^3} = 344 $
 $ {1^3} + {9^3} = 780 $
 $ {1^3} + {10^3} = 1001 $
 $ {1^3} + {11^3} = 1332 $
 $ {1^3} + {12^3} = 1729 $ or we can also write as $ {9^3} + {10^3} = 1729 $
Hence, 1729 natural numbers are there with the property that they can be expressed as the sum of the cubes of two natural numbers.
Problem fixing, repair, tuning, and knowledge acquisition can all be done by trial and error. The procedure is known as generate and test in the field of computer science (Brute force). When solving equations in elementary algebra, it's all about guessing and checking.
So, the correct answer is “1729”.

Note: The trial and error method works best with basic issues and games, and it is frequently utilised as a last resort when no obvious rule applies. This isn't to say that the strategy is necessarily irresponsible; an individual might be methodical in tweaking the variables in an attempt to sort through options that might lead to success.