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.
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Excellent Question
here i choose N=1 and discarded B,C,D then choose N=3 and Marked A as the option.
Nice One
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It seems like A is the answer whether N is even or odd. Are they just throwing in that N is odd to throw us off?

n = odd & least of 3 consecutive positive integers(n,n+1,n+2)

I took n as 3 and the other 2 numbers are 4 & 5

LCM of 3,4,5 = 60

Plug in 3 on each options and the answer should equal to LCM.

A. n(n+1)(n+2)
=>3.4.5
=>60

Hence A is the answer

let no be 1,2,3
LCM ; 1*2*3 ; 6
n=1
option A ; is valid

Plug in any value and check , let the 3 nos be 1 , 2 & 3

LCM of ( 1,2,3) = 6

Check the options -

(A) 1*2*3 = 6 (Possible)
(B) 1^3 + 3 = 4 (Not possible)
(C) 1(1^2 + 2) = 3 (Not possible)
(D) 1^2 + 3 = 4 (Not possible)
(E) 3 ( 1 + 1 ) = 6 (POssible)

Now check with

Here is why we need tp check all the options carefully as (E) will yeild the same result in this case of 1 ,2 & 3 and hence we need to plug in once again to confirm with 3,4 & 5 as ArjunJag1328
has done...

Correct Answer will definitely be (A)
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Testing x=1 will eliminate answers B,C,D but A and E will both produce the same result, so test x=3 to determine that A is correct.
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