What Is the Smallest Integer c for LCM(a, b, c) to Equal L?

[Shahed al mamun]

Recently I have faced the following problem. I'm given 3 integers a,b and L where LCM(a,b,c)=L and c is another integer. It is worthy of mentioning that the value of c will be smallest i.e if there are many possible values of c then we have to choose the one which is smallest.

I myself solved this problem partially using the following fact
LCM(a,b,c)=LCM(LCM(a,b),c)=L.Also I used the following procedure. Firstly,I found out LCM(a,b).Secondly, I divide L by LCM(a,b).

My procedure works for some cases.For example, if a=3,b=5 and L=30,the value of c will be 2 and my above procedure give correct result for this case. But there are some cases for which my procedure does't work.Here is a example. If a=10,b=15 and L=600 then the value of c will be 200 but my procedure give the value of c, 20. How can I get correct result for the given second case for which my procedure doesn't work?

 

[wabbit]

I myself solved this problem partially using the following fact
LCM(a,b,c)=LCM(LCM(a,b),c)=L.

Seems like a good start, you have now reduced the problem to an easier one.
Let's call that LCM(a,b), l
Now the problem is,
Given two integers l and L, find the smallest c such that LCM(l, c)=L
So how would you go about it ?