Question 847697
This isn't a formal proof, but it's just the basic outline (to get you thinking in the right direction)



If n^3 + 2 is odd, then n^3 must be odd. Why? Because the +2 makes you jump from even to even or from odd to odd. An example would be 7+2 = 9. Notice how we go from 7 to 9 when we increase by 2.


So we know n^3 is odd. It turns out that if you cube any even number, you get an even number. Similarly, if you cube any odd number, you get an odd number. So because n^3 is odd, this must mean that n has to be odd. 


We know n is odd. So n = 2k + 1 for some integer k. Square this to get n^2 = 4k^2 + 4k + 1 ---> n^2 = 2(2k^2 + 2k) + 1 ----> n^2 = 2m + 1 for some integer m (m = 2k^2 + 2k). This shows us that if n is odd, then n^2 is odd.



Let me know if this helps or not.