If $n$ is the least of 3 consecutive positive integers and $n$ is odd, what is the least common multiple of the 3 integers, in terms of $n$?

Thanks for the explanation. I got confused because I thought the LCM would just be the product, but since n is odd, n+2 is also odd, so they share a factor with n+1? Wait, n+1 is even. So n and n+2 are odd, so they don't share 2. But do they share any odd factor? Actually, n and n+2 differ by 2, so they could share a factor if that factor divides 2, but since they're odd, the only common factor is 1. So n and n+2 are coprime. But n+1 is even, so it shares a factor of 2 with... well, n and n+2 are odd, so n+1 shares no factor of 2 with them. But could n+1 share an odd factor with n? That would require the factor to divide n and n+1, so divide 1. So no. Similarly with n+2. So actually all three are pairwise coprime? Wait, n and n+1 are always coprime. n+1 and n+2 are always coprime. n and n+2: gcd(n, n+2) = gcd(n,2). Since n is odd, gcd(n,2)=1. So yes, pairwise coprime. So LCM = product. But the answer choices don't have product? Wait, choice A is n(n+1)(n+2). That is the product. So why is that not the answer? Maybe I'm missing something. Let me re-read the question: 'least common multiple of the 3 integers, in terms of n'. If they are pairwise coprime, LCM is product. So A. But then why is this a 505-555 question? Seems too easy. Maybe there's a trick: the integers are n, n+1, n+2. n is odd. So n+1 is even. Could n and n+2 share a factor? No, as argued. So A. But wait, the choices include n(n^2+2) which is n^3+2n. That's not the product. So maybe the LCM is not the product because they are not pairwise coprime? Let's test n=3: 3,4,5. LCM=60. Product=60. So A works. n=5: 5,6,7. LCM=210. Product=210. So A. So why is this a problem? Maybe the answer is not A? Could it be that the LCM is actually n(n+1)(n+2)/2? But that would be if they shared a factor of 2, but they don't because n and n+2 are odd. So no. So A seems correct. But then why is this in 505-555? Maybe it's a trick question where people think they need to divide by something. But actually, the answer is A. But wait, the choices: (C) n(n^2+2) = n^3+2n. That's not the product. So maybe the LCM is not the product? Let's check n=1: 1,2,3. LCM=6. Product=6. A works. n=7: 7,8,9. LCM=504. Product=504. So A. So I think A is correct. But the explanation might say something else? I'm confused. Maybe the question is asking for the LCM in terms of n, and the answer is A. So I'll go with A. But the difficulty is 505-555, so maybe it's easy. But I've seen harder problems. Anyway, thanks.

This one tripped me up because I assumed the LCM would be the product divided by something, but since n is odd, n and n+2 are both odd, so they don't share a factor with n+1? Wait, n+1 is even, so it shares a factor of 2 with... nothing because the others are odd. But could n+1 share an odd factor with n? No, because consecutive integers are coprime. So all three are pairwise coprime. So LCM = product. So A. But then why is this a 505-555? Maybe it's a trick to see if you realize they are coprime. Anyway, good question.

I got C initially because I thought the LCM might simplify, but then I tested n=3: numbers 3,4,5. LCM=60. A gives 3*4*5=60. C gives 3*(9+2)=33. So A is correct. So the answer is A. But the question is a bit easy for 505-555, maybe it's a lower difficulty.