2024/054) Find x and y such that $LCM(x,y) = LCM(12x,5y) = 720$

 As $LCM(x,y) = 720$ we need to find prime factors of 720

$720=2^4 * 3^2 *5$

So $x=2^a3^b5^c\cdots(1)$

And $y=2^d3^e4^f\cdots(2)$

Where $max(a,d) = 4\cdots(3)$

  $max(b,e) = 2\cdots(4)$

 $max(c,f) = 1\cdots(5)$'

Now

$LCM(12x,5y) = LCM(2^{a+2}3^{b+1}5^c $$, 2^d3^e5^{f+1})= 2^4 * 3^2 *5$

So $max(a+2,d) = 4\cdots(6)$

$max(b+1,e) = 2\cdots(7)$

$max(c,f+1) = 1\cdots(8)$

From (3) and (6) we get $d=4$ and $a \le 2$

From (4) and (7) we get $e=2$ and $b\le 2$

from (5) nd (8) we get $c=1$ and $f=0$

So we get following pairs $(2^0 * 3^0 * 5, 2^4*3^2)$ that is $(5,144)$

$(2^1 * 3^0 * 5, 2^4*3^2)$ that is $(10,144)$

$(2^2 * 3^0 * 5, 2^4*3^2)$ that is $(20,144)$

$(2^0 * 3^1 * 5, 2^4*3^2)$ that is $(15,144)$

$(2^1 * 3^1 * 5, 2^4*3^2)$ that is $(30,144)$

$(2^2 * 3^1 * 5, 2^4*3^2)$ that is $(60,144)$