# Math Help - lcm of k,k+1,k+2 where k≡3(mod 4)

1. ## lcm of k,k+1,k+2 where k≡3(mod 4)

Find the least common multiple of k,k+1, and k+2 where k≡3(mod 4).

Some ideas:
k≡3(mod 4) => k=3+4z for some z E Z
So k is odd, k+1 is even, and k+2 is odd.
Also, this question asks about the lcm of THREE numbers, so I don't think the the Euclidean algorithm (which only gives the gcd or lcm of TWO numbers) will help us in this question.

So how can we solve this?
Thanks for any help!

[also under discussion in math links forum]

2. Originally Posted by kingwinner
Find the least common multiple of k,k+1, and k+2 where k≡3(mod 4).

Some ideas:
k≡3(mod 4) => k=3+4z for some z E Z
So k is odd, k+1 is even, and k+2 is odd.
Also, this question asks about the lcm of THREE numbers, so I don't think the the Euclidean algorithm (which only gives the gcd or lcm of TWO numbers) will help us in this question.

So how can we solve this?
Thanks for any help!
Hint: $\text{lcm}\left(a,b,c\right)=\text{lcm}\left(\text {lcm}(a,b),c\right)$

3. Originally Posted by Drexel28
Hint: $\text{lcm}\left(a,b,c\right)=\text{lcm}\left(\text {lcm}(a,b),c\right)$
Is this always true? How can we prove it?

4. Since k is odd, k+1 is even, and k+2 is odd
then
GCD(k,k+1,k+2)=1 => LCM(k,k+1,k+2)=k(k+1)(k+2)

5. Originally Posted by dooping
Since k is odd, k+1 is even, and k+2 is odd
then
GCD(k,k+1,k+2)=1 => LCM(k,k+1,k+2)=k(k+1)(k+2)
How do we actually formally prove these two claims?