Question

Find the common factors of
(a) 4, 8 and 12
(b) 5, 15 and 25

Answer 1

Hint: We solve this problem by using the prime factorization method on each and every number. After the prime factorization then we take the common primes that are common for all three numbers given to find the required factors.

Complete step-by-step solution
The prime factorization method is nothing but writing the given number as the product of prime numbers starting from 2.
(a) 4, 8 and 12
Let us use the prime factorization method for number 4
We know that the number 4 is the square of 2
So, we can write the number 4 as
\[\Rightarrow 4={{2}^{2}}\]
Now, by using the prime factorization method to number 8 we get
\[\Rightarrow 8={{2}^{3}}\]
Similarly, by using the prime factorization method to 12 we get
\[\Rightarrow 12=2\times 6\]
Here, we can see that the above number is not yet reduced to the product of primes
By reducing the number 6 as a product of prime numbers we get
\[\Rightarrow 12={{2}^{2}}\times 3\]
Here we can see that the common product of primes that are present in three numbers 4, 8, and 12 is \[{{2}^{2}}\]
Let us write the common product of three numbers as
\[\Rightarrow C=1\times {{2}^{2}}\]
Now, we know that the factors of numbers 4, 8, and 12 are selecting the possible numbers from the common product in prime factorization.
Therefore, we can conclude that the common factors are 1, 2, and 4.
(b) 5, 15 and 25
Let us use the prime factorization method for the above numbers.
We already know that the number 5 is a prime number.
So, we can write the number 5 as
\[\Rightarrow 5=1\times 5\]
Now, by using the prime factorization method to number 15 we get
\[\Rightarrow 15=3\times 5\]
Similarly by using the prime factorization method to number 25 we get
\[\Rightarrow 25={{5}^{2}}\]
Here we can see that the common product of primes that are present in three numbers 5, 15, and 25 is \[5\]
Let us write the common product of three numbers as
\[\Rightarrow C=1\times 5\]
Now, we know that the factors of numbers 5, 15, and 25 are selecting the possible numbers from the common product in prime factorization.
Therefore, we can conclude that the common factors are 1 and 5.

Note: Students may make mistakes in the common factors. Common factors are the numbers that divide the given number exactly. Here, we need to mention all possible common factors. Here, we have that the common product of primes that are present in three numbers 4, 8, and 12 is \[{{2}^{2}}\] then the possible factors will be 1, 2, and 4. But students may mention only 4 because they misunderstand the common factors with HCF.