How do you find the LCM and HCF?

[chwala]

Homework Statement

Find the lcm and hcf for the numbers $1, 0.5 and 0.75$

Relevant Equations

lcm and hcf

as per my thinking, lcm of $1$ and $0.5$ is 1. And lcm of therefore, $1$ and $0.75$ should be $0.75$
from research,
lcm= lcm(numerator)/hcf(denominator) = 3/1
hcf=hcf(numerator)/lcm(denominator)= 1/4

i would appreciate more insight on this...

 

[Mark44]

Homework Statement: Find the lcm and hcf for the numbers $1, 0.5 and 0.75$
Homework Equations: lcm and hcf

as per my thinking, lcm of $1$ and $0.5$ is 1. And lcm of therefore, $1$ and $0.75$ should be $0.75$

No, the LCM (least common multiple) of 1, 1/2, and 3/4 is the smallest number that all three numbers divide evenly. Since 1 divides any number, it can be ignored in this example.
What's the smallest number that is divisible by 1/2 and 3/4?

Aside: In my experience, LCM and HCF (highest common factor) are more often used with integers, not fractions. A more usual problem would be to find the LCM of 2, 4, and 10, which would be 20. The LCM has to have two factors of 2 so that it's divisible by 4 (as well as 2), and a factor of 5. So the LCM of 2, 4, and 10 would be 20.

The HCF (highest common factor, also known as the greatest common factor) is the largest number that divides the numbers in question. For example, find the greatest common factor of 36 and 60.
$36 = 2^2 \cdot 3^2$ and $60 = 2^2 \cdot 3\cdot 5$
Therefore, the HCF (or GCF is $2^2 \cdot 3$, or 12.

from research,
lcm= lcm(numerator)/hcf(denominator) = 3/1
hcf=hcf(numerator)/lcm(denominator)= 1/4

i would appreciate more insight on this...

 

[chwala]

i am conversant with finding lcm and hcf of integers...now going back to this question would it be right to say now lcm $(0.5,0.75)$ = lcm(numerator)/hcf(denominator)=3/2 =$1.5$
hcf$(0.5,0.75)$=hcf(numerator)/lcm(denominator= 1/4 = $0.25$

 

[Mark44]

Yes to both.

 

[Rajiv00123]

The basic to find HCF and LCM with Example with using prime factorization method

 

[member 587159]

So you are asking to find least common multiple in the field of rational numbers $\mathbb{Q}$? Let me tell you something: this concept does not make a lot of sense in this context! That is because a least common multiple is determined uniquely up to unit and in the rational numbers all non-zero numbers are units. Put another way, the least common multiple of two non-zero elements will always be $1$ (up to unit).

More generally, in any field, the greatest common divisor of two non-zero elements will always be $1$.

 

[mathwonk]

The OP seems to have made an unstated assumption that by multiples, one means integral multiples. E.g. the smallest number that is an integral multiple of all of 1, .5, and .75, is 3. similarly one rational number r is assumed to divide another one s, for the purpose of this discussion, if s/r is an integer. In this sense one can perhaps make sense of this question. Mathqed is if course correct that when one considers rational numbers, the usual meaning of "divides" is that the quotient is a rational number. Then there is little interest to the question since then any non zero rational number divides any other.

Here the ancient text by Euclid is of interest since there the concept "divides", was phrased geometrically as "measures". So then .5 measures 1.5, because three copies of a length of .5 can be used to exactly measure a length of 1.5. So one can ask for the shortest length that can be measured by any of the lengths 1, .5, and .75, and the answer is a length of 3. Similarly the greatest length that can be used to measure any of them is .25.

 

[chwala]

The OP seems to have made an unstated assumption that by multiples, one means integral multiples. E.g. the smallest number that is an integral multiple of all of 1, .5, and .75, is 3. similarly one rational number r is assumed to divide another one s, for the purpose of this discussion, if s/r is an integer. In this sense one can perhaps make sense of this question. Mathqed is if course correct that when one considers rational numbers, the usual meaning of "divides" is that the quotient is a rational number. Then there is little interest to the question since then any non zero rational number divides any other.

Here the ancient text by Euclid is of interest since there the concept "divides", was phrased geometrically as "measures". So then .5 measures 1.5, because three copies of a length of .5 can be used to exactly measure a length of 1.5. So one can ask for the shortest length that can be measured by any of the lengths 1, .5, and .75, and the answer is a length of 3. Similarly the greatest length that can be used to measure any of them is .25.

so the terms lcm and hcf refer to what kind of numbers? Can they be used in regards to finding say, rational numbers, negative integers? or are they only restricted in this sense to natural numbers?

 

[jbriggs444]

so the terms lcm and hcf refer to what kind of numbers? Can they be used in regards to finding say, rational numbers, negative integers? or are they only restricted in this sense to natural numbers?

In the sense described by @mathwonk, I would say arbitrary real numbers. For instance, the HCF of ${\frac{3\pi}{2}}$ and $-2\pi$ is $\frac{\pi}{2}$ and their LCM is $6\pi$.

Meanwhile, the HCF and LCM of $1$ and $\pi$ is undefined -- no [non-zero] integer multiple or fraction of one can ever equal any [non-zero] integer multiple or fraction of the other.

The "HCF" would be the Highest member in the set whose members are integer Fractions of both numbers. The "LCM" would be the Least [strictly positive] member in the set whose members are integer Multiples of both numbers. For an empty set, the Highest or Least member is undefined.

 

[WWGD]

So you are asking to find least common multiple in the field of rational numbers $\mathbb{Q}$? Let me tell you something: this concept does not make a lot of sense in this context! That is because a least common multiple is determined uniquely up to unit and in the rational numbers all non-zero numbers are units. Put another way, the least common multiple of two non-zero elements will always be $1$ (up to unit).

More generally, in any field, the greatest common divisor of two non-zero elements will always be $1$.

Not sure what you mean in the last paragraph. Isn't gcd(2,4)=2?

 

[member 587159]

Not sure what you mean in the last paragraph. Isn't gcd(2,4)=2?

Yes. But if you consider the gcd over a field extension of the integers then the gcd is also 1.

 

[mathwonk]

In reference to post #8, comparable ("commensurable") numbers?, i.e. ones such that some integer multiple of one equals some integer multiple of the other.

 

[WWGD]

Yes. But if you consider the gcd over a field extension of the integers then the gcd is also 1.

Ring extension?

 

[chwala]

Ring extension?

Is this member still with us? assuming that you expect a response ...

 

[pbuk]

Ring extension?

Ring what extension? I don't get any answer on 587159