$\left[G:H\cap K\right]=\left[G:H\right]\left[G:K\right]$ if $\left[G:H\right],\left[G:K\right]$ are coprime

Let $G$ be a finite group and $H<G, K<G$.

I have shown that $\left[G:H\cap K\right]\leq\left[G:H\right]\left[G:K\right]$

But I do not know where to begin to prove the equality in case these indexes are coprime.

I'd appreciate it much if a hint could be given.

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Two hints:

1) $[G: H \cap K] = [G:H][H: H \cap K] = [G:K][K: H \cap K]$.
2) if $a \vert bc$ and $gcd(a,b) = 1$, then $a \vert c$.

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That was VERY helpful.. thank you very much –  dankilman Nov 27 '11 at 17:05
I would first show that $[G:H \cap K] = [G:H] \cdot [H : H\cap K]$.