Find x,y, and z such that x³+y³+z³=k, for each k from 1 to 100.
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First, we have to restrict x, y, and z to integers otherwise there are infinite combinations satisfying the given equation.
Next, since is greater than 100 and is less than 100, we have that .
Thus, the number of possible combinations of (x, y, z) satisfying the given expression is given by the permutation of 4 different objects taking 3 at a time where repetition is allowed given by possible combinations.
Examples, (x, y, z) = (1, 1, 1), (1, 2, 3), (3, 2, 4), etc satisfies the given equation.
However, cases of combinations of more than one 4 or 4 and 3s does not satisfy the equation. That is, (4, 4, 4), (4, 3, 3), (3, 4, 3), (3, 3, 4), (4, 4, 2), (4, 2, 4), (2, 4, 4), (4, 4, 1), (4, 1, 4), (1, 4, 4) does not satisfy the given equation.
Therefore, there are 64 – 10 = 54 possible combinations of values of x, y,, and z satisfying the given equation.