Question

The number of common prime factors 60, 75, and 105 are:
A. 2
B. 3
C. 4
D. 5

Answer 1

Hint: Here, we have to find the number of common prime factors of 60, 75, and 105. For that, we will first prime factorize all the three given numbers separately and then we will write their prime factors. Then we will see which of the factors are common to all three of them and then we will count the number of common prime factors. Hence, we will get our answer.

Complete step-by-step solution:
We here have to find the number of common prime factors of 60, 75, and 105.
For this, we will prime factorize all these three numbers separately.
We will first prime factorize 60. It is done as follows:
$\begin{align}
  & 2\left| \!{\underline {\,
  60 \,}} \right. \\
 & 2\left| \!{\underline {\,
  30 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Thus, we can write 60 as:
$60=2\times 2\times 3\times 5$
Hence, 60 has 3 prime factors- 2, 3 and 5.
Now, we will prime factorize 75. It is done as follows:
$\begin{align}
  & 3\left| \!{\underline {\,
  75 \,}} \right. \\
 & 5\left| \!{\underline {\,
  25 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Thus, we can write 70 as:
$70=3\times 5\times 5$
Hence, 70 has 2 prime factors- 3 and 5.
So far we can see that 60 and 75 has two common prime factors- 3 and 5. Now we will prime factorize 105 to see which of these two common prime factors of 60 and 75 is common with 105 too.
Prime factorization of 105 is shown as follows:
$\begin{align}
  & 3\left| \!{\underline {\,
  105 \,}} \right. \\
 & 5\left| \!{\underline {\,
  35 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Thus, we can write 105 as:
$105=3\times 5\times 7$
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 60 and 75.
Thus, we can say that 60, 75 and 105 have 2 common prime factors, namely, 3 and 5.
Hence, option (A) is the correct option.

Note: We also factorize two or more numbers together but it won’t work in this question as we need the exact amount of prime factors of all the three numbers here to compare and see the common factors. Thus, we prime factorize the numbers separately in this question to obtain the right answer.