Find three distinct positive integers $x,y,z$ such that $15xyz=\text{lcm}(x,y,z)\cdot \gcd(x,y,z)$

Question

Find three distinct positive integers $x,y,z$ such that $xyz=\text15\cdot{lcm}(x,y,z)\cdot \gcd(x,y,z)$

I've proved this for two numbers $x$ and $y$, but I'm not sure how to expand this to three numbers or even to $n$ numbers

Answer 1

Consider example, which will show that $xyz\ge lcm(x,y,z)\cdot gcm(x,y,z)$ .

Let $x=24$, $y=50$, $z=70$.

Then consider prime factorization of these numbers:

\begin{array}{|c|c|c|c|c|c|} \hline number & 2 & 3 & 5 & 7 \\ \hline x & 2^3 & 3 & - & - \\ y & 2 & - & 5^2 & - \\ x & 2 & - & 5 & 7 \\ \hline gcd(x,y,z) & 2 & - & - & - \\ \hline lcm(x,y,z) & 2^3 & 3 & 5^2 &7 \\ \hline xyz & 2^3\cdot 2 \cdot 2 & 3 & 5^2\cdot 5 & 7 \\ \hline \end{array}

GCD collects smallest powers of each prime.

LCM collects largest powers of each prime.

So, if consider any prime number $p$, then

$$x = p^a \cdot x_1,$$ $$y = p^b \cdot y_1,$$ $$z = p^c \cdot z_1,$$ where $a\ge 0, b\ge 0, c\ge 0$; then $$gcd(x,y,z) = p^{\min\{a,b,c\}}g_1;$$ $$lcm(x,y,z) = p^{\max\{a,b,c\}}l_1;$$ $$xyz = p^{a+b+c}m_1,$$ where $x_1,y_1,z_1$, $g_1$, $l_1$ and $m_1$ are not divisible by $p$.

But $a+b+c\ge \min\{a,b,c\}+\max\{a+b+c\}$. Since $a+b+c = \min\{a,b,c\}+medium\{a,b,c\}+\max\{a,b,c\}.$

And such property $-$ for each prime number from the table.

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If $15$ would be on RHS, then one can construct such $(x,y,z)$: $(3,5,15)$, $(5,15,30)$ for example.

How to find such triples? Let $$x=2^{a_1}\cdot 3^{a_2}\cdot 5^{a_3}\cdot 7^{a_4}\cdot \ldots;$$ $$y=2^{b_1}\cdot 3^{b_2}\cdot 5^{b_3}\cdot 7^{b_4}\cdot \ldots;$$ $$z=2^{c_1}\cdot 3^{c_2}\cdot 5^{c_3}\cdot 7^{c_4}\cdot \ldots;$$ where $a_1,b_1,...,c_4, ...$ $-$ all numbers are non-negative integer numbers.

Then $$xyz = 2^{a_1+b_1+c_1}\cdot 3^{a_2+b_2+c_2}\cdot 5^{a_3+b_3+c_3}\cdot 7^{a_4+b_4+c_4}\cdot \ldots;$$ $$ gcd(x,y,z)\cdot lcm(x,y,z) = 2^{\min\{a_1,b_1,c_1\}+\max\{a_1,b_1,c_1\}}\\ \times 3^{\min\{a_2,b_2,c_2\}+\max\{a_2,b_2,c_2\}}\\ \times 5^{\min\{a_3,b_3,c_3\}+\max\{a_3,b_3,c_3\}}\\ \times 7^{\min\{a_4,b_4,c_4\}+\max\{a_4,b_4,c_4\}}\\ \times \ldots; $$ Since there is $15=3\cdot 5$ near $gcd\cdot lcm$, then looking at powers of $2$, then at powers of $3$ etc, we conclude: $$ \left\{ \begin{array}{l} a_1+b_1+c_1 = \min\{a_1,b_1,c_1\} + \max\{a_1,b_1,c_1\}; \\ a_2+b_2+c_2 = \min\{a_2,b_2,c_2\} + \max\{a_2,b_2,c_2\} \large{+ 1}; \\ a_3+b_3+c_3 = \min\{a_3,b_3,c_3\} + \max\{a_3,b_3,c_3\} \large{+ 1}; \\ a_4+b_4+c_4 = \min\{a_4,b_4,c_4\} + \max\{a_4,b_4,c_4\}; \\ \ldots \end{array} \right. $$ from this point we conclude that:

• medium number of $a_1,b_1,c_1$ is $0$ (then the smallest is $0$ too);
• medium number of $a_2,b_2,c_2$ is $1$ (then the smallest is $0$ or $1$);
• medium number of $a_3,b_3,c_3$ is $1$ (then the smallest is $0$ or $1$);
• medium number of $a_4,b_4,c_4$ is $0$ (then the smallest is $0$ too);
• ...

Let us construct such sets of powers:
$a_1=0,b_1=3,c_1=0$; (smallest is $0$, medium is $0$, largest is $3$)
$a_2=1,b_2=2,c_2=0$; (smallest is $0$, medium is $1$, largest is $2$)
$a_3=2,b_3=1,c_3=1$; (smallest is $1$, medium is $1$, largest is $2$)
$a_4=1,b_4=0,c_4=0$; (smallest is $0$, medium is $0$, largest is $1$)
and let other primes have powers $0$ (for simplicity);

and finally compose $x,y,z$:

$x=2^{a_1}\cdot 3^{a_2}\cdot 5^{a_3}\cdot 7^{a_4} = 2^0\cdot 3^1\cdot 5^2\cdot 7^1 = 525$;

$y=2^{b_1}\cdot 3^{b_2}\cdot 5^{b_3}\cdot 7^{b_4} = 2^3\cdot 3^2\cdot 5^1\cdot 7^0 = 360$;

$z=2^{c_1}\cdot 3^{c_2}\cdot 5^{c_3}\cdot 7^{c_4} = 2^0\cdot 3^0\cdot 5^1\cdot 7^0 = 5$;

$gcd(x,y,z) = 5$, $lcm(x,y,z)=12\:600$, $xyz=945\:000$,
$945\:000 = 15\cdot 5 \cdot 12\:600$.

Just for training: reproduce this construction for getting triple $x=3,y=5,z=15$, or other one: $x=5,y=15,z=30$.