### [Aptitude] LCM, HCF, GCD: Basic concept, calculation, applications explained » Mrunal

To find LCM and HCF of (a/b) and (c/d) the generalized formula will be: H.C.F Is this answer still relevant and up to date? . multiples of 5 are 5,10,15,20, How to solve LCM-HCF related problems without mugging up any formulas? LCM of two numbers (56, 96); LCM of three numbers: (12,15,20); LCM of HCF of co-prime numbers (12,25); HCF vs LCM: #1 multiplication; HCF vs .. 1 of 3) · [ Reasoning] Calendar Questions: Finding day or date, concepts. voyancegeni.us's LCM calculator to find what is Least Common Multiple for group of whole numbers or integers 20, 25 and is the LCM for above group of.

Recall that LCM of numbers is their product when they do not have any common factor other 1. Therefore, LCM of the numerators 8, 7 and 3 is: Therefore, 1 is the common factor of the denominators 5, 4 and 8.

Therefore, LCM of the fractions: Find the least number which when divided by 3, 4 and 5 leaves a constant remainder 2 in each case. First find the LCM of the three numbers 3, 4, and 5. Since 3, 4, and 5 do not have any common factor, their LCM is their product. Now the required number is: Cross-check by dividing 62 by each of the given divisors: You will note that 62 leaves a constant remainder 2 on being separately divided by each of the given divisors 3, 4 and 5.

Find a smallest number that is divisible by 20, 25 and Also, the number is a perfect square. A number divisible by all the three numbers 20, 25 and 30 must their LCM.

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So, the LCM of 20, 25 and Find the minimum number of oranges in a bag that can be exactly divided among a group of 5 boys, a group of 7 girls and a group of 10 women? The required minimum number of oranges is the LCM of the three numbers: Since 5 is a factor of 10, so find LCM of only 7 and Again, since 7 and 10 do not have any common factortherefore, LCM of 7 and 10 is their product: It is a lattice because every pair of elements has a least upper bound, and greatest lower bound; and it is complete because the entire set has a least upper bound, 0, and a greatest lower bound, 1.

### How to Find the LCM of 8 & 12 - Video & Lesson Transcript | voyancegeni.us

I suspect that the definition you give is a useful version that works well in most cases; however, perhaps the definition I gave is a way that the LCM can be more general. Could you give me an example of why your definition might be more useful? Also the proofs that I have seen for existence of the LCM use the definition that I gave, and I'm yet to find one that uses the least positive number definition, though I wouldn't be surprised if one exists.

Part ii of the definition is particularly useful in proving existence. Could you show me to an existence proof which uses your definition? This is definitely helping me think about my course and consider the reasons for these definitions more carefully! I cannot help you at the level you are discussing, that of generalizing the LCM from its original, more intuitive application.

I will invite other math doctors to discuss this matter with you. Your definition is absolutely correct; and, unlike the "simpler" definition, it works in any commutative ring although, in general, LCMs need not exist or be unique.

The definition is the dual of the general definition for GCD. Note that the original article was about LCM a,b where either a or b is 0; this is not the case in your example with LCM 3,2.

In fact, I would say that there is no problem in considering that 0 is a common multiple of a pair of integers: The point is that it is not the least such multiple, where "least" must be understood with respect to the partial ordering induced by divisibility, since this is the meaning used implicitly in "least common multiple.

In any case, I think you know all this. The last question is why we have the "simpler" definition.

## [Aptitude] LCM, HCF, GCD: Basic concept, calculation, applications explained

It is already taught in elementary school, at a time where the general definition would be out of reach. Even at that stage, the definition can be very useful in practical applications like adding fractions ; the same is true for greatest common divisor GCD.

For many people, that is about all the mathematics they will need. I think that, if you want to learn math seriously, you have first to unlearn many of the false or incomplete things you were taught in school, because it was not possible at that time to give you strictly correct and complete definitions.

Please feel free to write back if you require further assistance. I'd like to tie things up with a comment on the big picture. It is quite common for a concept to start with a simple idea and a "naive" definition, and later be generalized.

## How to Find the LCM of 8 & 12

In this particular case, as has been mentioned, both definitions MUST continue in use in different contexts, because only the naive initial definition is understandable by most people who need the concept, while only the sophisticated general definition applies to cases beyond natural numbers. Most online sources, including Wikipedia and MathWorld, give the definition applicable to natural numbers. This is the appropriate definition for use as the Least Common Denominator of fractions, since denominators can't be zero.

It also fits the name: This definition is undoubtedly the source of the entire concept. Some sources give that same definition, but then add that if one of the numbers is zero, the LCM is 0 with or without explaining why this extension makes sense. This is probably the answer that should have been given to the original question from a computer science context, just looking for a reasonable value to give in this case. This elementary definition had to be extended in order to cover other contexts, as you noted.

As Doctor Rick pointed out, simply applying the basic definition to such cases would not work.

### Least Common Multiple(LCM)

The definition you are using is the result of a search for an appropriate extension, and is based on a theorem that is true in the natural number context, and turns out to be usable as a definition in the general case. It certainly would not be appropriate to start with this as a definition in elementary grades, but it would be possible to make the transition before getting to abstract algebra -- though it would never be particularly helpful in understanding the concept in its everyday applications.

I had trouble searching for a source for your definition, since the elementary definition is overwhelmingly common. As I mentioned, Wikipedia gives the elementary definition, but adds https: However, some authors define LCM a,0 as 0 for all a, which is the result of taking the LCM to be the least upper bound in the lattice of divisibility. This last thought leads to your definition, which is given at the bottom of the page in defining the lattice of divisibility: The least common multiple can be defined generally over commutative rings as follows: Let a and b be elements of a commutative ring R.

A common multiple of a and b is an element m of R such that both a and b divide m i. A least common multiple of a and b is a common multiple that is minimal in the sense that for any other common multiple n of a and b, m divides n.

In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are associates. In a unique factorization domain, any two elements have a least common multiple. In a principal ideal domain, the least common multiple of a and b can be characterised as a generator of the intersection of the ideals generated by a and b the intersection of a collection of ideals is always an ideal.

So, summing things up, the definition Doctor Rick gave is very useful in the everyday world; yours is useful in higher-level mathematics -- and both can peacefully coexist because they give the same result where both apply. Both are "the correct definition" within their own context.