If n is the least of 3 consecutive positive integers and

Two consecutive integers are co-prime, meaning that they do not share any common factor but 1. For example 4 and 5 are consecutive integers and they do not share any common factor but 1. This means that the leas common multiple of two consecutive integers is always their product: LCM(4, 5)=20.

So, for n, n+1 and n+2 each pair consecutive terms is co-prime (n and n+1, as well as n+1 and n+2). Now, since n is odd, then n and n+2 are also co-prime (consecutive odd integers are also co-prime: 3 and 5, 5 and 7, 7 and 9, ...), which means that all 3 integers are co-prime to each other, hence the least common multiple of the 3 integers is n(n+1)(n+2).

Answer: A.

Else you can pick some number for n, say 3. Then three consecutive integers will be 3, 4, and 5, their LCM is 3*4*5=60. Now, just plug 3 in the answer choices to see which yields 60: only A.

Note that for plug-in method it might happen that for some particular number(s) more than one option may give "correct" answer. In this case just pick some other numbers and check again these "correct" options only. For example if you pick n=1 then you get two "correct" choices A and E.

Hope it helps.