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

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SUMMARY

The discussion focuses on determining the smallest integer c such that LCM(a, b, c) equals a given integer L. The user initially applied the formula LCM(a, b, c) = LCM(LCM(a, b), c) = L, successfully solving some cases but encountering issues with others. For instance, with a=10, b=15, and L=600, the correct value of c is 200, while the user's method incorrectly suggested 20. The key takeaway is that the problem can be simplified to finding the smallest c such that LCM(l, c) = L, where l is LCM(a, b).

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Shahed al mamun
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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?
 
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Shahed al mamun said:
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 ?
 

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