Inspired by Sandeep Bhardwaj sir
− 6 π , π
Find the lowest common multiple (LCM) of the two numbers above.
To clarify, the LCM of two irrational numbers exists if and only if their ratio is rational.
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2 solutions
Then,-6Pi is also LCM, due to that Pi and 6Pi divides to -6Pi, and -6Pi divides to 6Pi
LCM According to this LCM exists for non - zero integrals!!
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Yup, with that definition, I agree...
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So , how and why are we talking about irrational numbers' LCM??
@Calvin Lin @Sravanth Chebrolu @Kalpok Guha @Guillermo Templado Please help me out!!
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@Ankit Kumar Jain – I learnt LCM for integers numbers, and it could be negative and positive, but I don't make rules and definitions on Brilliant...
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@Guillermo Templado – It depends on how you generalize the LCM.
Taking it from the positive integers to the integers, we define it as the smallest positive integer that is a multiple of both terms (else is 0). This would be uniquely defined.
Taking it from the integers to the reals, we define it as the smallest positive real that is a multiple of both terms (else is 0). This would be uniquely defined.
This is the much more natural initial extension, where we understand LCM and GCD using the Lattice structure of divisibility.
There is another extension over commutative rings, in which 2 elements could have multiple LCM's, that differ up to a multiplicative unit. Taken with this interpretation over the reals, we do say that both the positive and the negative values are LCM's.
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@Calvin Lin – Sir , I couldn't understand the last bit (your second paragraph) - 'There is another extension ...'.
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@Ankit Kumar Jain – I did not want to delve into ring theory . Unless you know commutative rings, principle ideal domains, generators, etc, that paragraph would not be easily explained / understood.
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@Calvin Lin – I am not familiar with ring theory and commutative rings.!!
Thanks!!
Is it a well known convention that LCM is positive?
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Yes in the context of real numbers.
In the context of abstract algebra rings, LCM has a different definition.
Kalpok but how could you ignore -ve sign it will change the number
Computing the LCM implies finding the smallest positive integer k s.t − 6 π k = p and π k = q where p,q ∈ Z . From here, we have that π k − 6 π k = q p ⟹ − 6 π π = q p . Since the LCM needs to be positive, i.e k ∈ Z + , therefore p = -1 and q = 6. Substituting these back into the original fractions we obtain k = 6 π .
L.C.M is always positive so here we ignore the negative sign.
π is common factor between both the numbers π , 6 π
The L.C.M of 1 and 6 is 6 .
From the above statements we can say the L.C.M is 6 π