A natural number n is said to be good if n is the sum of r consecutive positive integers, for some r 2 . Find the number of good numbers in the set 1,2, , 100) .

Step-by-step solution

All odd numbers are good because, 2 K+1=K+(K+1) All even numbers which have an odd factor greater than 1 are also good because, array r 2^n (2 K+1)=K+ (K 1)+(K 2)+ + (K (2^n-1 ) ) _ (2^n-1 ) terms + (K+2^n ) array So, only 2^n are the numbers which are not good. Numbers which are not good are 2^0, 2^1, 2^2, 2^3, 2^4 , 2^5 and 2^6 . Remaining 93 numbers are all good.