A highest common factor (HCF) is the largest factor that goes into multiple numbers. This can be represented as gcd(a,b)=c; where a and b are constants and c is the HCF. The purpose of finding an HCF is to simplify fractions; many fractions have similar numbers in its numerator (top part) and denominator (bottom part). Since we can eliminate numbers and variable that are in both the numerator and denominator it is useful to know the HCF so when we divide both parts by that number we get the most simplified version of that fraction.

15/21=(5*3)/(7*3)=5/7

From the fraction listed above one can recognize that after simplifying it into its prime factors there is a 3 in the numerator and denominator. Since anything divided by itself is 1 ex: 3/3=1we can eliminate the 3 from both parts of the fraction and we are left with 5/7. From this we would label 3 as the HCF or gcd(15,21) = 3.

490/147 = (7*7*5*2)/(7*7*3) = (5*2)/3=10/3

From this fraction we notice that both parts of the fraction have two 7s. This means that we can eliminate both of those 7s and we are left with 10/3. Since we were able to cancel off two 7s our HCF is 7*7 or 49. When trying to find a HCF one is most likely to run into this type when there is multiple factors combined together that can be cancled off; which means that it is good to keep in mind that you break everything down into prime numbers because some numbers are not as easy to find. Ex: 391s factors are 23 and 17.

While there are some HCFs that are not very common some are very easily recognizable. Ex: 600/400=3/2; gcf(600,400) = 200. When looking at this fraction it may be easy to notice 200 goes into both of them so there is no need to go into great detail. Writing the factors of 200 out (2*2*2*5*5) can get pretty long and daunting but if a pattern is easily recognized why not just take it out; that of course is the reasoning for creating the term HCF.

6/2=(3*2)/(2)=3/1=3

From this fraction we see that after canceling out our HCF (2) there is only 1 left in the denominator; since anything divided by 1 is itself than the fraction becomes a whole number.

2/6=(2)/(3*2)=1/3

This fraction looks very similar to the last because it is just its reciprocal. After eliminating the HCF there is only a 1 left in the numerator. This is completely fine; it is just good to recognize patterns and notice that fractions of this kind are just the reciprocal of a whole number.

(x*y^2*9)/(x^3*y*6)=(x*y*y*3*3)/(x*x*x*y*3*2)=(y*3)/(x^2*2)

This fraction is just like any other; the only difference is instead of some numbers there are variables. In algebra one learns that a variable is a number that can change and be influenced by other variables but it still is just like any other number. Just as every other problem we will find what is in both parts of the fraction and divide by the HCF. In this case the gcf(x*y^2*9, x^3*y*6) = x*y*3.