2
$\begingroup$

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with the least common multiple, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it. So anyway, here the problem goes:

Find all solutions in positive integers for the equation $$x-y^4= LCM(x, y).$$

Any subtantial answers or help would be truly greatly appreciated. Thanks so much :)

$\endgroup$
6
  • 2
    $\begingroup$ What is $LCM(1)$ supposed to mean...? $\endgroup$
    – CiaPan
    Commented Jul 23, 2015 at 21:56
  • 2
    $\begingroup$ You have $\operatorname{lcm}(x,1) = x$, so setting $y = 1$ doesn't work. $\endgroup$ Commented Jul 23, 2015 at 21:56
  • $\begingroup$ @DanielFischer I am thoroughly confused. I don't know how to do it. $\endgroup$
    – anonymous
    Commented Jul 23, 2015 at 21:58
  • $\begingroup$ You must have $x \mid x - y^4$ and $y \mid x-y^4$. That gives you some conditions to work with. $\endgroup$ Commented Jul 23, 2015 at 21:58
  • $\begingroup$ @DanielFischer Sorry I don't understand. Could you please explain in answer? $\endgroup$
    – anonymous
    Commented Jul 23, 2015 at 22:01

1 Answer 1

2
$\begingroup$

AFAIK the least multiple of any positive integer $x$ is $x$ itself. As $y^4$ is greater than zero, $x-y^4$ is less than $x$, so it can't be a multiple of $x$ (whether common with $y$ or not...). Of course that holds unless you allow ZERO as a multiple, in which case $x-y^4 = 0$ would give solutions.

$\endgroup$
5
  • $\begingroup$ What is AFAIK?? $\endgroup$
    – anonymous
    Commented Jul 23, 2015 at 22:04
  • $\begingroup$ It's 'As Far As I Know', AFAIR. $\endgroup$
    – CiaPan
    Commented Jul 23, 2015 at 23:06
  • $\begingroup$ Oh, and AFAIR is 'As Far As I Remember', IIRC. $\endgroup$
    – CiaPan
    Commented Jul 23, 2015 at 23:07
  • $\begingroup$ What is IIRC then? $\endgroup$
    – anonymous
    Commented Jul 24, 2015 at 10:44
  • $\begingroup$ STFN :) $\endgroup$
    – CiaPan
    Commented Jul 24, 2015 at 18:45

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .