Highest Common Factor or GCF. Properties of HCF

What are the factors of 40?

There are a total of eight factors of 40 viz. 1, 2, 4, 5, 8, 10, 20 and 40.
Why?
∵ these all numbers give the remainder zero when 40 is divided by 1, 2, 4, 5, 8, 10, 20 and 40.
How?
$\frac{40}{1}=40$
$\frac{40}{2}=20$
$\frac{40}{4}=10$
$\frac{40}{5}=8$
$\frac{40}{8}=5$
$\frac{40}{10}=4$
$\frac{40}{20}=2$
$\frac{40}{40}=1$

Let’s understand it by taking the two numbers as 4 and 10 and further, finding out their common factors.

Step1: find out the factors of 4 and 10 separately.
Factors of 4:
$\frac{4}{1}=4$
$\frac{4}{2}=2$
$\frac{4}{4}=1$
There are a total of three factors of 4 viz. 1, 2 and 4.
Factors of 10:
$\frac{10}{1}=10$
$\frac{10}{2}=5$
$\frac{10}{5}=2$
$\frac{10}{10}=1$
There are a total of four factors of 10 viz. 1, 2, 5 and 10.
Step2: Find out the common factors of 4 and 10.
Finally, we can say 1 and 2 are the common factors of 4 and 10, because the factors 1 and 2 do exist for both numbers 4 and 10.

Let’s learn HCF as an example.

Find the HCF of 12 and 20.
So, what are the steps?
Step 1: Find out the factors for 12.
Step 2: Find out the factors for 20.
Step 3: Find out the common factors of 12 and 20.
Step 4: Find out the greatest number among those common factors, that will be the HCF of 12 and 20.
Step1: Find out the factors for 12.
$\frac{12}{1}=12$
$\frac{12}{2}=6$
$\frac{12}{3}=4$
$\frac{12}{4}=3$
$\frac{12}{6}=2$
$\frac{12}{12}=1$
There are a total of six factors of 12 viz. 1, 2, 3, 4, 6 and 12.
Step2: Find out the factors for 20.
$\frac{20}{1}=20$
$\frac{20}{2}=10$
$\frac{20}{4}=5$
$\frac{20}{5}=4$
$\frac{20}{10}=2$
$\frac{20}{20}=1$
There are a total of six factors of 20 viz. 1, 2, 4, 5, 10 and 20.
Step3: Find out the common factors of 12 and 20.
∴ the common factors are 1, 2 and 4.
Step4: Find out the greatest number among those common factors, that would be the HCF of 12 and 20.
∴ the greatest number among the common factors 1, 2 and 4 is 4.
Hence, HCF or GCF or GCD of 12 and 20 is 4.

6 is a perfect number

Why?
∵ sum of factors of 6 is 12
and 6 × 2 = 12
How?
Step1: Find out the factors of 6.
$\frac{6}{1}=6$
$\frac{6}{2}=3$
$\frac{6}{3}=2$
$\frac{6}{6}=1$
Factors of 6 are 1, 2, 3 and 6
Step2: Find the sum of factors
1 + 2 + 3 + 6 = 12
∴ we can see, the sum of factors of 6 which is 12 is equal to twice the number itself.

HCF of two numbers 20 and 50 is 10.
It means the HCF 10 also divides the two numbers 20 and 50.

HCF of two numbers 27 and 60 is 3.
It means the HCF 3 can’t be greater than the two numbers 27 and 60

HCF of two numbers 27 and 54 is 27.
Here, 2 and 27 are factors of number 54. Also, 27 number is the number for which HCF will be calculated.
Therefore, HCF will be the smallest number of 27 and 54, which is 27.

HCF of 14 and 17 is 1.
Because, 14 and 17 are coprimes numbers.

HCF of 19 and 20 is 1.
Because, 19 and 20 are consecutive numbers.