
Hi Abhishek
1. First convert the number from base5 to base10 and then find the remainder. N = (232323...._{100 digits})_{5} = (2*5^{99} + 3*5^{98} + 2*5^{97} + 3*5^{96} + .... _{100 terms})_{10} ≡ (2 + 3 + 2 + 3 + .... _{100 terms}) mod 4 ≡ 250 mod 4 ≡ 2 mod 4.
Now N^{4231} ≡ 2^{4231} mod 4 ≡ 0 mod 4.
See that power of N here is in base5 but when it'll be converted to base10, it'll remain more than 1 and hence the number will always be divisible by 4.
2. There seems to be some problem with this question as if you consider all exterior angles to be equal then the polygon becomes a regular polygon and total integral possible values of n becomes 22 which is not in the options.
Only thing you need to know and use here is that Sum of all exterior angles of a planar polygon is 360 degrees.
Kamal Lohia
