Determine the LCM of the following pairs of integers:
(a) [45, 81]
(b) [2000, 2015]
(c) [1000, 1001]
(a) We use the prime decompositions of 45 and 81.
$45 = 3^2 \times 5^1$
$81 = 3^4$
So the LCM $[45,81] = 3^4 \times 5^1 = 405$.
(b) We use the prime decompositions of 2000 and 2015.
$2000 = 2^4 \times 5^3$
$2015 = 5^1 \times 13^1 \times 31^1$
So the LCM $[2000, 2015] = 2^4 \times 5^3 \times 13^1 \times 31^1 = 806000$.
(c) We use the prime decompositions of 1000 and 1001.
$1000 = 2^3 \times 5^3$
$1001 = 7^1 \times 11^1 \times 13^1$
So the LCM $[1000,10001] = 2^3 \times 5^3 \times 7^1 \times 11^1 \times 13^1 = 1001000$.