Question

Find the number of positive integers <1000 that can be expressed as 2^k−2^m, where k and m are non-negative integers.

Answer 1

My way of solving......
Let k=m+1
2^(m+1)-2^m <1000
2^m <1000
m<=9

therefore k<=10

For distinct value of k and m,
when k=1,
m=0
when k=2,
m=0,1
when k=3,
m=0,1,2
.
.
.
when k=10
2^k−2^m<1000
2^10-2^m<1000
m>=4
no of possible value of m is 5

total possible combination of (k,m)=50

For k=m, no solution because 0 is not a positive integer

total possible combination of (k,m)=50

Answer 2

Is it correct?