valid or invalid argument - contradicting arguments My professor asked us to assess the validity of the following argument:

Some rational numbers are powers of 5. All integers are rational. Therefore, some integers are powers of 5.

My professor went back and forth on the validity of the argument after I questioned his logic. Finally, he asserted that the argument is invalid and he gave this argument:

Now let us abstract this argument, letting R represent the set of rational numbers, P the set of powers of five, and G the set of integers.  Then the premises and conclusion become:
   1)  Some element of R belongs to P.
  2)  All elements of G belong to R.
    Conclusion:   Some element of G belongs to P.
As a model of this argument, let R = { a, b, c };  P = { c };  and G = { a, b }.
  Are the premises true?          1)  Some element of R belongs to P.
                                  2)  All elements of G belong to R.
  Does the conclusion now follow:  Some element of G belongs to P?

I've spoken to another professor who says the argument is valid and I've seen different answers to this problem including this one:

Link here:
see exercise 6
So I ask, what is the truth? 
EDIT: 
In addition, for any open sentence P(x), is
$$\exists x \in \mathbb{Q} P(x)$$ 
not equivalent to,
$$\exists x (x \in \mathbb{Q} \Rightarrow P(x))$$
and if these are not equivalent, then why does the author, in the example below, rephrase the following quantified statement as an implication? 

EDIT 2: 
This post here is similar to my last question.
thank you.
 A: The argument is invalid. Here is a refutation by logical analogy:
Some coins are dimes
All nickels are coins
Therefore, some nickels are dimes
The argument based on formal logic notation fails, since it uses the wrong symbolizations. For example, some rational numbers are powers of five needs to be symbolized as:
$$\exists x (Q(x) \land R(x))$$
and not as:
$$\exists x (Q(x) \rightarrow R(x))$$
So ... either the text was asking you to find the error in the 'Solution' ... or the text provided a horribly mistaken Solution! Given how everything else labeled 'Solution' seems to be treated as the actual answerk to the exercises, I fear it's the latter .. what text is this?!
A: The argument:
"Some rational numbers are powers of 5. All integers are rational. Therefore, some integers are powers of 5."
is not valid
The rationals which are powers of 5 are not necessarily integers. For example $\frac {1}{32}$ is a power of 5 but it is not an integer. All integers are rational does not mean all rationals are integers.
If all rationals were powers of 5 then we could argue that all integers are powers  of 5, but from some rationals are powers  of 5 we can not argue that some integers are powers of 5.   
