Question

The number of positive integers with the property that they can be expressed as the sum of cubes of two positive integers in two different ways is?
(a) 1
(b) 100
(c) Infinite
(d) 0

Answer 1

Hint: We will write an equation representing the question statement. Then we will multiply the numbers in the equation by another number. This will tell us if the question statement is true or not. That will lead us to finding the number of such positive integers.

Complete step-by-step answer:
We have to express one number as the sum of cubes of two positive integers in two different ways. We know some examples like $1729={{10}^{3}}+{{9}^{3}}={{12}^{3}}+{{1}^{3}}$ and $4104={{2}^{3}}+{{16}^{3}}={{9}^{3}}+{{15}^{3}}$.
Let $a$ and $b$ be one pair and $x$ and $y$ be another pair such that ${{a}^{3}}+{{b}^{3}}={{x}^{3}}+{{y}^{3}}$ and all four numbers are not equal to each other. Let the sum of their cubes be denoted by $p$.
So we have $p={{a}^{3}}+{{b}^{3}}={{x}^{3}}+{{y}^{3}}$. Now, let $n\ne 0$. We will multiply the entire equation by the number ${{n}^{3}}$. We get the following equation, ${{n}^{3}}p={{(na)}^{3}}+{{(nb)}^{3}}={{(nx)}^{3}}+{{(ny)}^{3}}$.
As $n$ can take infinitely many values, there will be an infinite number of positive integers with the property that they can be expressed as the sum of cubes of two positive integers in two different ways.

So, the correct answer is “Option c”.

Note: The main part in this question is to realize the existence of the number $n$. We eliminated option (a) and (d) as soon as we came up with two examples of the said property. Eliminating option (b) is a difficult task. Hence, we chose to prove that there have to be infinite such numbers.