If a positive integer is divisible by 45 it must be divisible by
both 9 and 5.  Since any integer of the form XY45 ends in 5 it 
is automatically divisible by 5. Therefore we only need to make 
sure the number is divisible by 9 as well.  A positive integer is
divisible by 9 if and only if the sum of its digits is a multiple
of 9.  Since the sum of the last two digits is a multiple of 9,
the sum of the first two digits must also be a multiple of 9.  
Therefore, the only possibilities are

1845, 2745, 3645, 4545, 5445, 6345, 7245, 8145, 9045, and 9945.

So there are exactly 10 such positive integers.

Edwin