Question

Can we find LCM of following :

A) irrational and irrational

B) irrational and rational

Why or why not?

Answer 1

Dear student,
According theory of LCM and HCF to find the LCM or HCF of a and b we must write those numbers as product of powers of prime factors.

Then to find HCF raise the common prime factor to the lesser of the powers in which it appears in prime factors product.

Product of the above numbers is HCF.

Ex: 175,230

175=5^2*7^1

230=2^1*5^1*23^1

To find the HCF take the prime factors to the least power present in each product

Here 5^1 is the prime factor with least power common to each number. so HCF=5

similarly to find the LCM raise the each of the prime factor to highest of the powers in which it appears in the product.

Here Prime factors we have 2,5,7,23

LCM=2^1*5^2*7^1*23^1=8050

a*b=175*230=40250 which is equal to LCM*HCF=5*8050=40250

But we cannot write irrational numbers as product of prime factors .

Hence we cant find LCM and HCF if irrational numbers present. LCM and HCF are defined for natural numbers only .
Let x and y be positive real numbers. Then N is the least common multiple of x and y if N/x and N/y are both integers and no smaller positive number has this property.

With 5*sqrt(2) and 3*sqrt(2) their least common multiple is 15*sqrt(2), because it's the smallest number that's an integer multiple of both.

However, they don't always have an LCM. Take sqrt(2) and sqrt(3). There is no number L such that L/sqrt(2) and L/sqrt(3) are integers, otherwise their quotient, sqrt(2/3), would be rational. And it isn't.

If x and y are irrational, they have an lcm iff x/y is rational. And the lcm is x multiplied by the denominator of x/y in simplest form.
Hope you help this....