Since 

60=2*2*3*5, and 24=2*2*2*3

The product of an even number and 60 is divisible by 24, for such a 
product gets an additional 2 factor from the even number multipled by 60.  
However the product of an odd number and 60 does not get an additional 
2 factor from the odd number multiplied by 60, and thus is not divisible 
by 24, but only by 60. So the sequence of such integers are 

1*60, 3*60, 5*60, ... , 15*60

17*60 is 1020, more than 1000, so these 15 are the only products of an
odd number and 60 which is less than 1000, and thus the only integers 
less than 1000 which are multiples of 60 but not of 24. 

So the answer is 15.

Edwin