Question

write the set of all integers whose cube is an even integer?

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Answer 1

You have 2 cases. n1=2k for some integer k, and n2 = 2j+1 for some integer j. If you take n1, and cube it, you get 8k^3 which is even, so all evens are in that set. Now n2 squared is 8j^3 + 12j^2 + 6j + 1= 2(4j^3 +6j^2 +3j) +1 which is odd, so no odd numbers have even cube. Thus the answer is all even numbers.

Answer 2

As said the answer is going to be an integer

so it can be either positive or negative it doesn't matter

as we know the basic rule of algebra that odd into odd gives odd and even into even gives even

so the all the values are going to be in form of

(2n)^3 where n belongs to I and we can't give or tell all the integers as there are infinite integers

so the answer will always be in form of (2n)^3 where n belongs to I ( integers)