Question

Find all of the common prime factors of 30 and 105.
1) 2 and 3
2) 3 and 5
3) 5 and 7
4) 3 and 7

Answer 1

Hint: The number of common prime factors of 30 and 105 must be determined here. To do so, we'll prime factorize each of the three integers separately, then write down their prime factors. Then we'll calculate the number of common prime factors by determining which factors are shared by both of them. As a result, we will receive our response.

Complete step-by-step solution:
Before solving this problem, we need to understand the concept of prime factorization that is: Prime factorization is the process of dividing a number into its prime factors, which are all prime numbers. In other terms, it deconstructs numbers into their primary components.
Now, we come to the problem, that is
We here have to find the number of common prime factors of 30 and 105.
For this, we will prime factorize all these two numbers separately.
We will first prime factorize 30. It is done as follows:
First, we need to divide by 2 we get:
\[2\left| \!{\underline {\,
  30 \,}} \right. =15\]
Now, in next step we have to divide by 3 because 15 is divisible by 3 we get:
\[3\left| \!{\underline {\,
  15 \,}} \right. =5\]
Now, we have to divide by 5 because 15 is divisible by 3 we get:
\[5\left| \!{\underline {\,
  5 \,}} \right. =1\]
In the next step we need to again divide by 1 we get:
\[1\left| \!{\underline {\,
  1 \,}} \right. =1\]
\[30=2\times 3\times 5\]
Hence, 30 has 3 prime factors- 2, 3 and 5.
Now, we will prime factorize 105. It is done as follows:
First, we need to divide by 3 because 105 is divisible by 3 we get:
\[3\left| \!{\underline {\,
  105 \,}} \right. =35\]
then we again divide by 5 again, we get:
\[5\left| \!{\underline {\,
  35 \,}} \right. =7\]
Now, in next step we have to divide by 3 because 15 is divisible by 3 we get:
\[7\left| \!{\underline {\,
  7 \,}} \right. =1\]
In the next step we need to again divide by 1 we get:
\[1\left| \!{\underline {\,
  1 \,}} \right. =1\]
Thus, we can write 105 as:
Hence, 105 has 3 prime factors- 3, 5 and 7.
Now, we can see that out of these 3 factors of 105, 3 and 5 are common with that of 30.
Hence, option (B) is the correct option.

Note: We can also factor two or more numbers together, but this won't work in this case since we need the precise number of prime factors for all three numbers to compare and see what they have in common. To get the correct answer, we prime factorize the numbers independently in this question.