GCD and LCM number theory problem I am having some trouble finishing a solution to this problem, so I thought I might ask you all for help.
The problem goes as follows:
Find all natural numbers $a,b&gt;0$ for which holds:
$$lcm(x,y)-gcd(x,y)=\frac{xy}{2021}$$
The problem is annoted with the hint, that 2021=43*47.
I have tried using the fact, that the $gcd=d$ from which follows $x=a*d$ and $y=b*d$. The $lcm(x,y)$ therefore must be $lcm(x,y)=a*b*d$.
Given this and the fact that the product of two numbers $x,y$ and their $gcd$ and $lcm$ is the same, meaning
$$xy=lcm(x,y)*gcd(x,y)=abd*d$$ I arrive at the equation:
$$(abd)-d=(abd^2)/2021$$
which equals
$$2021(ab-1)=abd$$
Where do I go from here? Any help or correction of mistakes that I may have made is greatly appreciated!
You rewrite it as $(2021 - d)ab = 2021$. Now use the factorization of $2021$ given in the hint.
Hint: write it as $\,f(a)=0\,$ so $\,a\mid f(0)\,$ since $\,f\,$ is a polynomial in $\,a\,$ with integer coef's. Thus the integer values that $a$ can take are constrained to the finite number of factors of $\,f(0)\,$ (if $\,f(0)\neq 0)$
Remark $ $ This can be viewed as special (trivial) case of the ubiquitous Rational Root Test, i,e., an integer root of $f$ divides its constant term.