Find the number of positive integers not exceeding 1000 that

Find the number of positive integers not exceeding 1000 that are either the square or the cube of an integer.

Solution

let us consider call S the set of those squares, and C the set of those cubes.

We must find |S C|.

There are 31 squares in A={1,2,….,1000} which are 12, 22, 32,…., 312. (Observe that 322 >1000). Thus |S|=31. There are exactly 10 cubes in A, which are 13, 23,,…., 103 and so |C|=10. Since | S C|= |S| + |C| | S C |, we only need to find |S C| to reach our goal. But out of the 10 cubes, only 1, 64 and 729 are squares (please, verify this statement with your calculator!), so | S C |= 3. Finally, |S C|= |S| +|C| |S C |= 31 + 10 –3 = 38. So there are 38 positive numbers not exceeding 1000 that are either the square or the cube of a positive integer.