# Can dividing two rational numbers yield an integer?

I wanted to know if two non-int numbers (non-zeroes) when divided with each other can give an integer or not.I believe that's a NO. However I know they can only yield an integer $1$ provided both are the same (i.e) $\frac{2.5}{2.5}$. Am I correct. Just wanted to be 100% sure. How about when they are multiplied ?

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You can certainly divide a rational by a rational and get an integer. Example: $\frac{2/7}{1/14}=4$. If we have two numbers, rational or not, such that $\frac{r}{s}$ and $\frac{s}{r}$ are both integers, then $r=\pm s$. –  André Nicolas Jul 27 '12 at 9:44
That is not what that wikipedia page says. Remember that integers are also rational numbers. –  Tobias Kildetoft Jul 27 '12 at 9:45
The Wikipedia article also doesn't say "no", what it actually says is that dividing one rational by another never produces something that isn't rational. So you can't get irrationals out of rational division. (Don't forget Raymond Manzoni's comment above at this point). –  Luke Mathieson Jul 27 '12 at 9:47
$\frac{2.4}{1.2}=2$, so the answer to your question is yes. –  Joel Reyes Noche Jul 27 '12 at 9:52
@MistyD : Its not the point of integers or rationals. Its the point of G.C.D . If the numerator and denominator have a G.C.D of 1, they wont yield integers and leave you with some decimal part –  Iyengar Jul 27 '12 at 9:53

Of course you can:

2.2/1.1 = 2

2.2 and 1.1 are both obviously rational non-integers, and 2 is an integer. Is this what you wanted?

You can even divide two irrational numbers and get an integer:

(2 * Pi) / Pi = 2

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See my earlier comment above. –  Joel Reyes Noche Jul 27 '12 at 12:37

Let $n$ be an integer and $p:=\frac{n}{n+1},q:=\frac{1}{n+1}$ then $p/q=n$. Note that both $p$ and $q$ are non-integers (as $\gcd (n,n+1)=1$).

Also if $r:=\frac{n^2}{n+1},s:=\frac{n+1}{n}$ then $rs=n$. Clearly both $r$ and $s$ are non-integers, because $\gcd (n^2,n+1)=\gcd (n,n+1)=1.$

So for every integer $n$ you can find non-integral rationals whose quotient is $n$ and non-integral rationals whose product is $n$.

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