A number theoretical problem

  • Thread starter Shahed al mamun
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In summary, the conversation discusses a problem where the smallest value of c needs to be found given 3 integers a, b, and L where LCM(a, b, c)=L. The speaker shares their procedure of using the fact that LCM(a, b, c)=LCM(LCM(a, b), c)=L and dividing L by LCM(a, b). However, it is mentioned that this procedure does not work for all cases. The conversation then shifts to finding the smallest value of c given two integers l and L where LCM(l, c)=L.
  • #1
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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  • #2
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 ?
 

1. What is a number theoretical problem?

A number theoretical problem is a mathematical problem that involves the study of integers and their properties. It often involves finding patterns and relationships between integers, and can be used to solve real-world problems such as encryption and data security.

2. What are some common number theoretical problems?

Some common number theoretical problems include the Goldbach conjecture, the Riemann hypothesis, and the Collatz conjecture. These problems have remained unsolved for many years and continue to intrigue mathematicians.

3. How do mathematicians approach number theoretical problems?

Mathematicians use a variety of techniques to approach number theoretical problems, including algebraic, analytic, and geometric methods. They also use computer algorithms and advanced mathematical tools to explore and analyze these problems.

4. Why are number theoretical problems important?

Number theoretical problems are important because they have practical applications in fields such as cryptography, data security, and coding theory. They also help us better understand the fundamental properties of numbers and their relationships.

5. Are there any practical implications of solving a number theoretical problem?

Yes, there can be practical implications of solving a number theoretical problem. For example, the proof of the Riemann hypothesis would have significant impacts on prime number theory and the distribution of prime numbers, which has practical applications in cryptography and data security.

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