BdMO National 2013: Higher Secondary 5
Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$.
- asif e elahi
- Posts:185
- Joined:Mon Aug 05, 2013 12:36 pm
- Location:Sylhet,Bangladesh
Re: BdMO National 2013: Higher Secondary 5
We prove that $x$ is prime.Let $x$ has a divisor $d$ and $x=dy$.So $x$ divides $dy$.This implies $x$ divides $d$ or $x$ divides $y$.But both of $d$ and $y$ are less than $x$.So $x$ is a prime.$x$ is divisible by only $1$ and $x$.So the ans is $2$.