The number of 5-tuples (a, b, c, d, e) of positive integers such that I. a, b, c, d, e are the measures of angles of a convex pentagon in degrees ; II. a b c d e ; III a, b, c, d , e are in arithmetic progression is

Step-by-step solution

a+b+c+d+e=540 Let say first term = a Common difference d n 2 [2 a +( n -1) d ]=540 a +2 ~d =108 Case-1 d =0 a =108 (108,108,108,108,108) Case-2 d =1(106,107,108,109,110) n 2 [2 a +( n -1) d ]=540 a +2 ~d =108 Case-1 d =0 a =108 (108,108,108,108,108) Case-2 d =1 (106,107,108,109,110) Similarly it goes up to d =35 for d >35 , interior angle >180^ which is not possible So form d =0 to d =35 total 36 tuples are possible