Is a number meeting these conditions divisible by forty-nine? I am not a mathematician, I'm a linguistics PhD student. As part of my research I need to put various convoluted sentences through various syntactic transformations and see then check whether people think they are true or not. Mathematical statements (well, some of them) suit my purposes very well, because they are less context dependent and can be straightforwardly assigned a truth value (i.e. be deemed true or false). The problem is that I'm not a mathematician. When these sentences get a bit convoluted, I have a bit of a problem knowing whether they are true or false myself (before they undergo various syntactic transformations). 

I have a particular sentence which states that if a given number is:


an integer
divisible by 7 (meaning it will yield an integer if divided by 7)
a square number


then it is divisible by 49. I intuitively believe this to be correct (although I can't explain why). Is this actually true? I don't want to waste everybody's time by starting with an untrue untransformed sentence.
You can derive this fairly directly from Euclid's Lemma, which says that if a product $a \cdot b$ is divisible by prime $p$, then either $a$ or $b$ (or both) is divisible by $p$.

So, $n^2$ divisble by 7 means $n \cdot n$ divisible by 7 which (by Euclid's Lemma) means $n$ divisible by 7 and thus $n^2$ divisible by 49.

Of course it took even Euclid a little work to prove the Lemma
suppose this number is x. 

Now we know:


x is an integer (Assumption: positive integer)
x is divisible by 7
x is a square number





From Point 1 we know its a positive integer. From that we can say x is 1 or greater than 1. (x=1 or x>1)
From Point 2 we know x is divisible by 7. That brings us to conclude that x is at least 7. Or multiple of 7 since its divisible by 7. We can say (x=7 or x=7*y) Here y can be any positive integer (y=1 or greater)
Now from Point 3 we know that its a square. that means it has some form of (x= 7^2) or (x=7^(2) .y) or (x=7^2 . z^2). where y = z^2. 




What I am trying to say here is 


7 is a prime number. 
x is a square and divisible by 7.


This forces the number x to be a product of 7^2. Which makes it divisible by 49. because 49 is 7^2.