Amit,
Here it goes(hope I dint add too much confusion here):
Two cases to consider here:
Case 1: x+x^3 will be divisible by 4 if x and x^3 individually are divisible by 4. If 2 components individiaully are divisible by a number then their sum definitely is
This means x is divisible by 2
Case 2: x and x^3 are both not divisible by 4 then x+x^3 will be divisible by 4 if x and x^3 remainders sum is divisible by 4
Whats unique about x^3 if I am not mistaken is that x will give the same remainder as x^3 when x^3 is not divisible by that number. Therefore their sum will never be diviisble by 4
Eg: x=3 x^3=27 when divided by 4 Both give a remainder of 3 so the sum of the remainders 6 is not divisble by 4 so x+x^3 is not divisible by 4
The 2nd CASE is not possible since x^3+x is given to be divisible by 4 so it has to be case 1
Hope this helps!
Someone feel free to correct me if I am mistaken in any of the statements made above.
I have tried to adapt the explanation to this problem (since we are dealing wiht the same number's different powers) but please see Ian's explanation to my question here:
https://www.beatthegmat.com/is-x-16-y-8- ... 26911.html
Regards,
Cramya
