The largest common factor of all the given numbers is known as the Highest Common Factor of the numbers.
The highest number can be divided exactly into two or more numbers without any remainders.
It is also known as the Greatest Common Divisor (GCD).
The easiest way to find the HCF of two or more given numbers is to create a factor tree.
In contrast to this, another term frequently discussed in arithmetic is LCM. It stands for Least Common Multiple. In mathematics, the least common multiple or LCM of two numbers X and Y is denoted by the symbol LCM (X, Y). The LCM is the smallest or least positive integer divisible by the numbers we have taken. Let us now take the positive integers 4 and 6 to explain this concept to you.
Multiples of 4 include 4, 8, 12, 16, 20, 24, and so on.
Multiples of 6 include 6, 12, 18, 24, and so on.
From this, we can see that the common multiples of 4 and 6 are numbers such as 12, 24, and so on. Amongst all these numbers, the least common multiple of 4 and 6 is 12. This makes it the LCM of 4 and 6.
Here are a Few Highest Common Factor Examples :
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The above picture shows how you can calculate the H.C.F. of 90 and 30.
How to Calculate HSF(Highest Common Factor) of Any Two or More Given Numbers?
There are three methods of how to find the highest common factor of any two or more given numbers:
Factorization Method
Prime Factorization Method
Division Method
Let’s know how these methods work.
Factorization MethodIn the HCF by factorization method, we find the greatest common factor by listing down the factors of the numbers.Step 1: List down the factors of all the given numbers.Step 2: Look for the highest factor among the listed factors which are common for both the given numbers. Let us take an example:1. Problem: Find the H.C.F. of 36 and 45.Solution: 36 = 1,2,3,4,6.9,12,18,3645 =1,3,5,9,45Since the highest common factor of 36 and 45 is 9, therefore H.C.F of 36 and 45 is 9.
Prime Factorization Method(Also known as the Factor Tree Method) Let us consider an example,Step 1: In calculating the HCF by prime factorization, we factorize the numbers into prime numbers, which is known as the prime factors.Step 2: Start by dividing the given numbers by 2(the first prime number), go on dividing till you can’t divide the number any further.Step 3: Finally, then write the numbers as a product of the prime numbers. The product of these common factors is the highest common factor of the given numbers.Here’s an example for better understanding. Find the HCF by prime factorization of the numbers 24 and 36.Solution: To find the HCF by prime factorization,24 = 2×2×2×3×136 = 2×2×3×3×1Since the common factors of 24 and 36 are 2×3×2×1= 12Therefore, the HCF of 24 and 36 is 12.
Division MethodIn the division method, we follow the following steps:Step 1: Treat the smallest number as the divisor, and the bigger number as the dividend.Step 2: Further divide the divisor by the remainder.Step 3: Stop dividing when the remainder equals zero.Here’s an example,Find the H.C.F. of 12,18.Solution:Here, 6 is the H.C.F.
How to Find the HCF of 3 Numbers or How to Find the Highest Common Factor:
We can find the HCF of 3 numbers either by Prime Factorisation Method or by Division Method. However, the steps for finding the highest common factor remains the same as above.
Here are a few highest common factor examples of how to find the highest common factor,
1. Find the HCF of 3 numbers 15,30 and 90 using the Prime Factorization method.
Solution:
3 | 15 |
5 | 5 |
1 |
2 | 90 |
3 | 45 |
3 | 15 |
5 | 5 |
1 |
2 | 30 |
3 | 15 |
5 | 5 |
1 |
15 = 5×3×130= 5×3×2×190= 3×3×2×5×1. The common factors here are 1,3,5. Therefore, the highest common factor of the numbers,15,30 and 90 is 5×3×1=15.2.
2. Find the highest common factor of 184,230 and 276 by using the Division Method.
Solution:
46) 230 (6
-184
0 ) 184 (6
-184
The Common Factor of 184 and 230 is 46.
46) 276 (6
-276
The Highest Common Factor of 276 and 46 is 46. Therefore, the highest common factor(H.C.F.) of 184,230 and 276 is 46.
HCF and LCM Tricks
During an exam, students should know proper time management skills. These tips and tricks will help them save time all while allowing them to calculate answers faster.
The HCF of two given numbers can never be greater than any of the numbers.
The LCM of given numbers can never be less than any of the numbers.
The HCF of two or more prime numbers will always turn out to be 1. This is because prime numbers are only divisible by 1 and the number itself.
The LCM of two or more prime numbers is simply the product of those two numbers.
We use the formulas given below to calculate the HCF and LCM of fractions directly.
(a) LCM of Fractions =LCM of the Numerator/ HCF of the Denominator
(b) HCF of Fractions =HCF of the Numerator/ LCM of the Denominator
Solved Examples
Q1) What is the H.C.F. of 25,30,35 and 40?
Sol: Solving by Factorisation Method,
25 = 5×5
30 = 5×3×2
35 = 5×7
40 = 2×2×2×5
Therefore, the H.C.F of the numbers 25,30,35 and 40 is 5.
Q2) Find the highest common factor of 64 and 80 by using the prime factorization method?
Sol: Solving by Prime Factorization Method, The factors of 64 and 80 are,
64=2×2×2×2×2×2×1
80=2×2×2×2×5×1
Therefore, the Highest Common Factor of 64 and 80 is, 2×2×2×2×1=16
Q3) Find the highest common factor or the Greatest Common Divisor of 16 and 27.
Sol: Therefore, the Highest Common Factor of 16 and 27 is 1.
Q4) Find the Highest Common Factor (H.C.F.) of 12,15 and 45.
Sol: Solving the above-given question using the division Method,
12) 15 (1
-12
3)12(4
-12
We come to know that 3 is the Highest Common Factor of 12 and 15.
Now, let’s find the Highest Common Factor of 3 and 45.
3) 45 (15
-45
Therefore, the Highest Common Factor of the numbers 12,15 and 45 is 3.
Q5) Find the H.C.F. of 136,170 and 255 by using the division method.
Sol: Let us find the Highest Common factors of 136 and 170 first.
136) 170 (1
-136
34)136(4
-136
Highest Common Factor of 170 and 136 is 34.
Now, we need to find the H.C.F. of 34 and 255,
34) 255 (7
-238
17)34(2
-34
The H.C.F. of 34 and 255 is 17. Therefore, the required highest common factor of 136,170 and 255 is 17.
Q6) Find the H.C.F. of 160, 220, 340.
Sol: Let us solve the question by using,160 =2×2×2×2×2×5×1220 =2×2×5×11×1340 =2×2×5×17×1So, the numbers common in the sequence are 2×2×5×1. Therefore, the H.C.F of the numbers 160,220 and 340 is 20.
Important Terms You Need To Know:
Factors -A factor is a number that exactly divides another number without leaving any remainder. Example: 3 and 5 are factors of 15
Prime Factors-A prime factor is a number that does not have any factors except 1 and the number itself. Example: 2,3,5,7,11,13,17 and so on.
Factor Tree-
A diagram used to list down the prime factors of a number in a tree diagram.
Common Factor-
A factor is a common factor when it is a factor of two or more numbers.
Example: Let’s find out the common factor of 6 and 3.
First, list down the factors of both the given numbers.
6=2×3×1
3=3×1
3 is a factor of 6 as well as it is a factor of 3.
Hence, 3 is a common factor of 6 and 3.
Conclusion
We use the highest common factor to simplify fractions. The highest common factor or the greatest common divisor is the greatest number that divides exactly into two numbers. The article is helpful for students to practise problems on HCF.
