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 ?
Can dividing two rational numbers yield an integer?
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arithmetic
rational-numbers
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0What's three-fifths divided by three-tenths? – 2012-07-27
2 Answers
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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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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 $n$.