Do You Remember LCM?
2 , − 4
Find the lowest common multiple of the two numbers above.
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5 solutions
I'd like to address to Mr.Venkata. Why LCM for smallest positive integer? It says lowest common multiple and it freely can go toward negative infinity. In such case there is no lcm for 2 and -4.
Oleg Yovanovich
i think lcm(a,b) = a.b/gcd(a,b); so why it can not go in negative, it's primitive
LCM(a,b) for two integers a and b actually denotes the smallest positive integer that is divisible by both a and b. Hence, here LCM(2, (-4)) = 4.
LCM(a,b) for some
Wolfram Mathworld states that " LCM(a,b) is the smallest positive number m for which there exist positive integers n a and n b such that
n a a = n b b = m . "
Since there exists no positive integer n a for which − 4 will be positive, LCM(-4,2) does not exist by this definition.
When the question can only be solved by using one definition, the question maker should clarify what definition he is using.
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This is less a "math problem" than it is a survey question on which definition of LCM you go by. Would love to see it deleted.
Yes, I do agree with your argument strongly. As per Wikipedia :-
In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple) of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b.
The Wikipedia definition is in accordance with the posed problem, although as you have observed, MathWorld has a different one. The definition should be mentioned in the problem.
See my report on this as I have faced the same problem with the definitions. These problems are really ambiguous !
The definition is given for positive integers, but it doesn't imply that L.C.M. is not defined for the numbers other than positive integers. So, it simply means that the definition didn't tell us anything about the numbers other than positive integers.
In favor of the above statement :
"0 is not a positive integer" doesn't imply that I'm saying that 0 is a negative integer.
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Nowhere in the definition does it say that a , b have to be positive integers. If the authors wanted their definition to hold for only positive numbers, they would've mentioned it.
In the absence of such a mention, it is implies that the authors wanted to include negative numbers , and that the LCM of negative numbers does not exist.
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@Siddhartha Srivastava – Why are you including that the LCM of negative numbers doesn't exist ??? LCM of negative numbers exist too.
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@Sandeep Bhardwaj – We define the LCM in a certain way. It doesn't "exist" naturally.
What I'm saying is that the LCM of negative numbers is left undefined by most definitions, much like 0 0 is left undefined.
The definition you are using is uncommon (at least to me). The site I linked to is a reputable site and also leaves the LCM of negative numbers undefined.
If you want to define the LCM for negative numbers, you should provide the definition you are using.
@Sandeep Bhardwaj – Sir, Siddharta is not saying that LCM does not exist for negative numbers, but the fact that there is no single definition of LCM . Depending on the usage of various definitions, the answers change. For example, the Wikipedia definition does not allow for irrationals in domain although they do involve rationals and integers, and MathWorld definition allows only for positive integers, but both sources being very reliable. I hope you understand the problem now , and sorry if this had been a trouble for you.
Why the answer is not 2? I almost trapped and still don't understand this. Lucky there is no 2 in the option hhee
0 could well be interpreted as being both positive and negative (positive nothing and negative nothing) as well as being neither positive nor positive.
Under this interpretation, the LCM of two co-prime numbers is 0 and their HCF is 1
Typo. It should Be That is Divisible by Instead Of That Divides
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Thanks, I have now corrected it :).
I think your confusion is clarified by solving this.
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I solved this question before solving yours. Although this question is almost alright (it would've been better if the author specified the definition to use) and I am not so sure about yours ! The problem finally is that there is no single definition of LCM , and depending on the definition you think is correct, the answers change !
Oh, I can now remembered, I answered it -2. It is only then that I realized that -2 is the GCF(greatest common factor) not the LCM.
May I know, what would be the L.C.M. of -2 and -4?
Some define multiples of positive integers or natural numbers only. With this interpretation, the LCM does not exist. Problems that depend on such variable semantic issues should make clear the interpretation that is being assumed.
If we go by this , won't the LCM of any two numbers be − ∞ ??
Just wondering . . . since when was the definition of LCM extended to include those of negative numbers?
I don't know abput the definition of LCM.
But usually for these kinds of questions I would look at 2=2 -4=-1 2 2=-1*2^2
So to my understanding the lower common multiple must contain all the factors, that are included in the 2 numbers. 2 is a part of 2^2 so it isn't counted separately. So we have to consider -1 and 2^2 so we get -1*2^2=-4 this is both divisible by 2 and by -4, therefor it does the job.
I have never heard of a restriction that negative numbers aren't allowed.
the final number simply has to contain the fators of the 2 numbers. At least thats how I've learned and used it until now
It is only a convention that the least common multiple is positive. What is the justification for that convention?
Multiples of 2 (2, 4, 6, 8, 10, ...)
Multiples of -4 (-4, 0, 4, 8, 12, ...)
LCM (2, -4) = 4
those are not the multiples of -4..... multiples of -4 (-4,-8,-12,-16...so on and so forth)
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I totally agree. You cannot find a common multiple of a positive and a negative number
I don't think lcm of a negative and a positive number is defined.lcm means least common multiple,and we start multiples from 1 and not any negative number.as such,there shouldn't be any lcm as -4 has all multiples negative and 2 has all multiples positive.
Multiples of -4 are 0, -4, 4, -8, 8, ..... Another convention is that "lowest value" means "closest to zero". In that case, the LCM of (2, -4) is either of -4 or 4. Perhaps this is the reasonable reason that the LCM is by a convention chosen to be positive.
good answer
∣ − 2 ∣ = 2 ∣ 4 ∣ = 4 = 2 2 ∴ L C M ( − 2 , 4 ) = 2 2 = 4
I emphatically agree with Karthik. Brilliant's dfn of LCM is lacking. LCM is actually the smallest positive integer that is divisible by both a and b
This is a bad question, because it doesn't rely on any mathematical understanding, but only on knowledge of a convention decided about an edge case, at one point in time, at one point in space. this question should be retired.