a collection s of integers is defined by the following three rules: (i) 2 is in s; (ii) for every x in s, 3x and x + 7 are also in s; (iii) no integers except those defined by rules (i) and (ii) are in s. what is the smallest integer greater that 2004 which is not is s?
The set S is closed under multiplication by 3 and addition by 7. This means that if an integer is in S, then its multiples and its 7-successors are also in S.
We can use this to prove that all multiples of 7 are in S. Indeed, 2 is in S, so 3 * 2 = 6 is also in S. Then 7 * 6 = 42 is in S, and so on. This shows that all multiples of 7 are in S.
Now, let's consider the integer 2004. It is not a multiple of 7, and it is not 7 more than a multiple of 7. So, if 2004 is in S, then there must be some integer x in S such that 2004 = 3x + 7. But this means that x = 668. However, 668 is not in S, because it is not 2 more than a multiple of 7.
Therefore, 2004 is not in S. The smallest integer greater than 2004 which is not in S is 2005.