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Question

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

A
1
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B
100
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C
Infinite
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D
0
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Solution

The correct option is A Infinite
Let n be the positive integer with the property that it can be expressed as the sum of the cubes of 2 positive integers in 2 different ways. Then n=a3+b3=c3+d3.
Then any number of the form m3n also can be expressed as sum of the cubes of two integers in two different ways, as m3n=(ma)3+(mb)3=(mc)3+(md)3. Since m can take infinitely may values there will be such numbers available.
Now make sure that there exist at least one such number n.
Let, n=1729=1728+1=1000+729
Since there is at least one such n, there will be infinitely many such numbers.

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