Question

The number of positive integers with the property that they can be expressed as the sum of the cubes of $2$ positive integers in two different ways is

• A. $10$
• B. $100$
• C. $40$
• D. $\infty$

Answer 1

The correct option is D

$\infty$

Explanation for the correct answer:

Find the required number of positive integers.

Assume that, $n$ is a positive integer with such that it can be expressed as the sum of the cubes of $2$ positive integers in two different ways.

Therefore, $n=a^3+b^3=p^3+q^3$.

Since any number $k^{3}n$ can be expressed as the sum of the cubes of $2$ positive integers in two different ways.

$k^{3}n={\left(ka\right)}^3+{\left(kb\right)}^3={\left(kp\right)}^3+{\left(kq\right)}^3$

Since there are infinitely many possible values of $k$.

Therefore, The number of positive integers with the property that they can be expressed as the sum of the cubes of $2$ positive integers in two different ways is infinite.

Hence, option $D$ is the correct answer.