Find the LCM of the following:
(a) $[60, 100]$
(b) $[600, 1 000]$
(c) $[6 000, 10 000]$
What do you notice about your results?
(a) The prime decompositions are
$60 = 2^2 \times 3^1 \times 5^1$
$100 = 2^2 \times 5^2$
The LCM $[60,100] = 2^2 \times 3^1 \times 5^2 = 300$
(b) The prime decompositions are
$600 = 2^3 \times 3^1 \times 5^2$
$1000 = 2^3 \times 5^3$
The LCM $[60,100] = 2^3 \times 3^1 \times 5^3 = 3000$
(a) The prime decompositions are
$6000 = 2^4 \times 3^1 \times 5^3$
$10000 = 2^4 \times 5^4$
The LCM $[60,100] = 2^4 \times 3^1 \times 5^4 = 30000$
We notice that
$$[10a,10b] = 10 \times [a,b]$$
for integers $a,b$.