To find largest positive integer K, that divides 81n+20n−1 for n∈N, we start evaluating the expression for small values of n For n=1:811+20.1−1=100 For n=2:812+20.2−1=6600 Now, The greatest common divisor of 100 and 6600 is 100 . Hence, K=100 is the largest integer that divides 81n+20n−1 for all n∈N and the divisior of 100 are 1,2,4,5,10,20,25,50,100 ⇒S=1+2+4+5+10+20+25+50+100=217 Hence, S−K=117