2
$\begingroup$

What is the sum of all positive integers $n$ that satisfy $$\mathop{\text{lcm}}[n,100] = \gcd(n,100)+450~?$$

This problem seems really interesting, any hints are greatly appreciated. I did prime factorization of 100 to try and gain some information.

$\endgroup$
1
  • $\begingroup$ try $100n=gcd(100,n)^2+450gcd(100,n)$ $\endgroup$
    – Dexpectra
    Nov 3, 2017 at 12:21

4 Answers 4

1
$\begingroup$

Hint:

gcd$(n,100)=$lcm$(n,100)-450$

If $p=$lcm$(n,100),$

As $100|p,$ gcd$(p,450)$ must divide $(100,450)=50$

$\implies50$ must divide $n,$ let $n=50m$

$50$gcd$(m,2)=50$lcm$(m,2)-450$

$\iff$gcd$(m,2)=$lcm$(m,2)-9$

Now gcd$(m,2)$ should be $1$ or $2$

$\endgroup$
2
  • $\begingroup$ I'm able to see that n=250 is a solution, but I'm not sure if I can find any other solutions. $\endgroup$
    – ddswsd
    Nov 3, 2017 at 12:34
  • $\begingroup$ @ddswsd, If $(m,2)=1;m$ is odd lcm$(m,2)=9+1\implies m|10\implies m=5$ Now if $(m,2)=1;m$ is even lcm$(m,2)=9+2$ which is odd, but lcm$(m,2)$ is even as $m$ is even $\endgroup$ Nov 3, 2017 at 12:37
1
$\begingroup$

Hint: Write $d=\gcd(n,100)$. Then $n=dn'$ and $\operatorname{lcm}(n,100)=100n'$. Therefore, $100n' = d + 450$ and so $d$ is a multiple of $\gcd(100,450)$.

Solution:

$d$ is a multiple of $50$ and also a divisor of $100$. Hence, $d=50$ or $d=100$. But $d$ cannot be $100$ because $450$ is a not a multiple of $100$. Thus, $d=50$ and $n'=5$, which gives $n=250$.

$\endgroup$
0
$\begingroup$

$\text{lcm}(n,100)=L;\;\text{gcd}(n,100)=G$

product of $lcm$ by $gcd$ is equal to the product of the two numbers

$LG=100n;\;L=G+450$

$G(G+450)=100n$

$G^2+450G-100n=0$

$G=-225+\sqrt{225^2+100n}$

$n=250;\;G=50;\;L=500$

No other solutions because the $GCD$ must be less than $100$

Hope this is useful

$\endgroup$
0
$\begingroup$

Hint: Suppose $n>100$. Multiply both side of the equation by $gcd(n,100)$ to get: $$ 100n = gcd(n,100)^2 +450 gcd(n,100) < 100^2 +100\cdot 450 \implies n<550 $$

Now, since we have $50 |450-lcm(n,100)=gcd(n,100)$, We can deduce that $50|n$. Since $lcm(n,100)>450$ we can rule out $50,100,150,200,300,400$ and we are left with the candidates $250,350,450,500$. Checking all the numbers yields that the only possible answer is 250.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.