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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 $a,b>0$ 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!

Bill Dubuque
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3 Answers3

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You rewrite it as $(2021 - d)ab = 2021$. Now use the factorization of $2021$ given in the hint.

WhatsUp
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  • I have been trying this approach, but am still not getting anything. Do I need to guess for which range numbers this could be true, or am I completely missing something? –  Mar 07 '21 at 11:48
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    The product of three integers is equal to $2021$. Since $2021 = 43 \times 47$, there are not many choices. Essentially there are only two possibilities: $1 \times 43 \times 47$ and $1\times 1\times 2021$. And you should consider different permutations of the factors. – WhatsUp Mar 07 '21 at 11:49
  • Wouldn't that leave no possible solutions? Given that problems like this normally do have at least one solution that makes me think I am getting something wrong. So can you give me a hint as to whether there are solutions and I am getting something wrong? –  Mar 07 '21 at 15:54
  • Why do you think there are no solutions? We may have e.g. $a = 43, b = 47, 2021 - d = 1$ (and some more). – WhatsUp Mar 07 '21 at 16:58
  • Oh yes you are right! I kept mistaking $a$ and $b$ for $x$ and $y$ and therefore wrongly thought that $d$ could not be bigger than $a$ and $b$. Thanks for the quick answers! –  Mar 07 '21 at 18:52
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Hint

Let $(x,y)=d$ and WLOG $x/a=y/b=d\implies(a,b)=1$

$$abd-d=\dfrac{abd^2}{2021}$$

$$\iff\dfrac{2021(ab-1)}{ab}=d$$

But $2021=43\cdot47$

So, $a,b\in[1,43,47,2021]$

Can you take it home from here?

lab bhattacharjee
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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.

Bill Dubuque
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