Finding HCF and LCM: A Quick Guide

[chaoseverlasting]

Homework Statement

This is slightly embarassing, but how do you find the hcf and lcm?
For example, 22,24,30. What would the hcf and lcm be?
Or,
The period of f(x)=sinx+cosx is the lcm of 2pi, 2pi, what would that be?

Homework Equations

HCF*LCM=f(x)*g(x)

 

[quantumdude]

I know what an LCM is, but what's an HCF? Did you perhaps mean GCF?

 

[HallsofIvy]

Perhaps that was "Highest" Common Factor. A friend of mine who is not "mathematically" inclined suggested I say "smallest common factor" rather than "least common factor". She said that always made her wonder- what factor is "least common"!

Chaoseverlasting, in an attempt to reduce some of the chaos, remember that "Least Common Multiple" means the smallest number that is a multiple of each. For example, the least common multiple of 6 and 4 is 12: 12 is 2(6) and 3(4)- it is a multiple of both and is the smallest number that has that property. The least common multiple of 6 and 3, on the other hand, is just 6: 6= 2(3) and 1(6).

Now, in general, what is the least common multiple of any number, a, and itself. What is the "greatest (or highest) common factor"- that is a number such that each of the given numbers is a multiple of IT- if both numbers are equal?

 

[chaoseverlasting]

Thanks, that helped.

 

[tim_lou]

know anything about euclidean algorithm?

for HCF I guess you mean highest common factor, which is usually called the GCD, Greatest Common Divisor.

let's say you want to find GCD of 45 and 12, then
GCD(45,12)=GCD(45-12,12)=GCD(45-12-12-12,12)=GCD(9,12)=GCD(9,12-9)=GCD(9,3)=GCD(9-3-3-3,3)=GCD(0,3)

obviously, GCD(45,12)=GCD(0,3)=3

now, for lcm,
LCM(a,b)=a*b/GCD(a,b)

 

[jing]

In the UK, HCF is what we use for what in the US you term GCD or GCF.