Find the number of positive integers <1000 that can be expressed as 2^kâ2^m, where k and m are non-negative integers.
To find the number of positive integers less than 1000 that can be expressed as 2^k - 2^m, we can analyze the possible combinations of values for k and m.
First, let's establish some limits for k and m. Since we are looking for positive integers less than 1000, we know that k and m must be non-negative integers. Additionally, the largest possible value for k is 9, because 2^10 = 1024, which is greater than 1000. We can also set an upper limit for m as 9, to avoid any negative results.
Next, note that 2^k - 2^m can be written as 2^m (2^(k-m) - 1). The expression 2^(k-m) - 1 will be a positive integer if and only if k-m > 0, which means k > m.
Now, let's go through all the possible combinations of values for k and m.
For m = 0, we have k > 0, which gives us 9 possible values for k (1 to 9).
For m = 1, we have k > 1, which gives us 8 possible values for k (2 to 9).
For m = 2, we have k > 2, which gives us 7 possible values for k (3 to 9).
And so on, until m = 9, which gives us k > 9, resulting in 0 possible values for k.
Therefore, the total number of positive integers less than 1000 that can be expressed as 2^k - 2^m is:
9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45
So there are 45 positive integers less than 1000 that can be expressed as 2^k - 2^m, where k and m are non-negative integers.