Determine the following:
(a) $[2, 3, 5, 7]$
(b) $[24, 35, 51, 64]$
(c) $[11, 121, 132, 99, 77]$
(a) Since all of 2,3,5,7 are pairwise co-prime, we have
$$[2,3,5,7] = 2 \times 3 \times 5 \times 7 = 210$$
(b) We perform the prime decompositions as follows
$ 24 = 2^3 \times 3^1 $
$ 35 = 5^1 \times 7^1 $
$ 51 = 3^1 \times 17^1 $
$ 64 = 2^6 $
The LCM is therefore
$$ 2^6 \times 3^1 \times 5^1 \times 7^1 \times 17^1 = 114240 $$
(c) We recognise that a common factor of the numbers is 11, and use $[ab,ac]=a[b,c]$ from exercise 7.
$$ [11, 121, 132, 99, 77] = 11 \times [1, 11, 12, 9, 7] $$
The prime decompositions of 11, 12, 9 and 7 are
$ 11 = 11^1 $
$ 12 = 2^2 \times 3^1 $
$ 9 = 3^2 $
$ 7 = 7^1$
So we have
$$\begin{align} [11, 121, 132, 99, 77] &= 11 \times [1, 11, 12, 9, 7] \\ \\ &= 11 \times 2^2 \times 3^2 \times 7^1 \times 11^1 \\ \\ & = 30492 \end{align}$$