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number system and geometry
by Abhishek TG - Monday, 29 September 2014, 07:44 PM
 
1.
The people of island named TG use the number system to base 5.
if N= 2323............23 upto 100 digits. find the remainder when N^4231 is divided by 4.

2
if the external angle in degrees, of a n sided polygon, are all integers , which are in AP.how many values are possible for n ?

Re: number system and geometry
by TG Team - Wednesday, 1 October 2014, 10:43 AM
  Hi Abhishek smile

1. First convert the number from base-5 to base-10 and then find the remainder.
N = (232323....100 digits)5 = (2*599 + 3*598 + 2*597 + 3*596 + .... 100 terms)10 ≡ (2 + 3 + 2 + 3 + .... 100 terms) mod 4 ≡ 250 mod 4 ≡ 2 mod 4.

Now N4231 ≡ 24231 mod 4 ≡ 0 mod 4. smile

See that power of N here is in base-5 but when it'll be converted to base-10, 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. smile

Kamal Lohia