HCF which means Highest Common Factor is also known as GCD (Greatest Common Divisor). There are various methods to find HCF. You probably want the quickest method to find HCF, but you have to go through the books to find the highest common divisor. So, now we know HCF full form i.e Highest Common Factor.
Hcf full form and various methods to find HCF
HCF is the smallest number that can divide two or more numbers simultaneously. There are several scientists and mathematicians who worked on the HCF. Mathematicians like Lehmar and Euclidean came with different algorithms to find the GCD. The approaches are different but the results are the same.
Four Methods To Find HCF
- Quick Difference Method
- Factorization Method
- Prime Factorization Method
- Division Method
Quick Difference Method To Find HCF
First, we like to go through the fastest method to find HCF. This method is applicable for any given sets of numbers. But, the working rule is so simple and it consists of mostly three steps.
Working Rule of The Quickest Method:
- Find The Least Difference of the sets of numbers
- If The Difference is a factor of the numbers, then it is the HCF you are looking for
- If not, then prime factorize the difference, then one of the numbers which is the factor is the HCF.
Example of Quickest Method To Solve HCF
Q: Find The HCF of 9,12
Step 1: Difference of 12-9=3
Since 3 is the factor of both 9 and 12.
Q: Find The HCF of (8,14)
Step1: Difference: 14-8=6
Here, 6 is not a factor of anyone
Step 2: Prime Factorisation of 6: 2*3
Now, 2 is the factor of both, and 3 is not a factor.
Q:Find The HCF of (15,30,40)
Step 1: Difference: 30-15=15, 40-30=10, 40-15= 25
Step 2: Lowest Difference: 10
Step 3: Prime Factorisation of 10: 2*5
Now, 5 is the factor of all three numbers and 2 is not a factor of all three.
So, HCF(15,30,40) = 5
Q: HCF of 8, 9, 25
Step 1: Differences: 9-8=1,25-9=16,25-8=17
Step 2: Smallest Difference: 1
So, HCF of 8, 9, 25 is 1
From these four examples, you can solve any HCF-related questions easily in 5 seconds approximately.
Factorization Method To HCF
Find the HCF of 12, 24, and 48.
Let us first list all the factors of each number.
Factors of 12 are 1, 2, 3, 4, 6 and 12
Factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24
Factors of 48 are 1, 2, 4, 6, 8, 12, 24, and 48
The common factors of 12, 24, and 48 are 1, 2, 3, 4, 6, and 12.
The highest common factor is 12.
H.C.F. by prime factorization method
Let us consider an example.
Find the H.C.F. of 12, 30, and 48.
First, we find the prime factors of 24, 36, and 48.
Method of H.C.F.
12 = 2 × 2 × 3
30 = 2 × 2 × 3 × 5
48 = 2 × 2 × 2 × 2 × 3
The common prime factors = 2, 2, 3
H.C.F. = 2 × 2 × 3 = 12
Step I: Treat the smallest number as a divisor and the bigger number as a dividend.
Step II: The remainder becomes the divisor and the divisor becomes the dividend.
Step III: Repeat this process till the remainder becomes zero. The last divisor is the H.C.F.
Properties of HCF (Highest Common Factor)
- If a divides the product b⋅c, and HCF(a, b) = d, then a/d divides c.
- If m is a non-negative integer, then HCF(m*a, m*b) = m*HCF(a, b).
- If m is any integer, then HCF(a + m*b, b) = HCF(a, b).
- If m is a positive common divisor of a and b, then HCF(a/m, b/m) = HCF(a, b)/m.
- The HCF is a commutative function: HCF(a, b) = HCF(b, a).
- The following versions of distributivity hold true:
- hcf(a, lcm(b, c)) = lcm(hcf(a, b), hcf(a, c))
- lcm(a, hcf(b, c)) = hcf(lcm(a, b), lcm(a, c)).
So, in this article, we discuss HCF full form and various methods to find HCF. If you want to tell us more about the quickest method to find HCF. Please write to us on [email protected]