2014/0041) If a,b≥3,prove that 2^b−1 does not divide 2^a+1

There are 2 cases
case 1
b is a factor of a say a = mb

then 2b−1 is a factor of 2mb−1 so 2b−1 is not a factor of 2a+1 as remainder = 2
case 2
b is not a factor of a so a = mb + c where c < b

2a+1=2mb+c+1
= 2c(2mb−1)+2c+1
now 2b devides 2c(2mb−1) but as c < b
2c+1<2b−1as2c+2<2c+1

so it does not devide