LCM from Prime Decomposition/Examples/39, 102 and 75

Example of Use of LCM from Prime Decomposition

The lowest common multiple of $39$, $102$ and $75$ is:

$\lcm \set {39, 102, 75} = 33 \, 150$

$\ds 39$

$=$

$\ds 3 \times 13$

$\ds 102$

$=$

$\ds 2 \times 3 \times 17$

$\ds 75$

$=$

$\ds 3 \times 5^2$

$\ds \leadsto \ \ $

$\ds 39$

$=$

$\ds 2^0 \times 3^1 \times 5^0 \times 13^1 \times 17^0$

$\ds 102$

$=$

$\ds 2^1 \times 3^1 \times 5^0 \times 13^0 \times 17^1$

$\ds 75$

$=$

$\ds 2^0 \times 3^1 \times 5^2 \times 13^0 \times 17^0$

$\ds \leadsto \ \ $

$\ds \lcm \set {39, 102, 75}$

$=$

$\ds 2^1 \times 3^1 \times 5^2 \times 13^1 \times 17^1$

$\ds $

$=$

$\ds 33 \, 150$

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: Exercise $11$