Factorial !

It equals $\large\ \frac { \displaystyle \prod _{ i=1 }^{ 2018 }{ lcm\left( 1, 2, ..., \left\lfloor \frac { 2019 }{ i } \right\rfloor \right) } }{\displaystyle \prod _{ i=1 }^{ 2018 }{ lcm\left( 1, 2, ..., \left\lfloor \frac { 2018 }{ i } \right\rfloor \right) } }$ So find where $lcm\left( 1, 2, ..., \left\lfloor \frac { 2019 }{ i } \right\rfloor\right) \ne lcm\left( 1, 2, ..., \left\lfloor \frac { 2018 }{ i } \right\rfloor\right)$ This only happens when $\left\lfloor \frac { 2019 }{ i } \right\rfloor\ne\left\lfloor \frac { 2018 }{ i } \right\rfloor$ That is when $i$ is a divisor of 2019,and $\dfrac { 2019 }{ i }$ can be expressed as $p^n$ ,where $p$ is a prime, $n$ is a positive integer.

$2019=673\times3$ ,solving and get $i=673$ or $3$ .Putting back, $673\times3=2019$