Question

How many prime factors does 120 have?

Answer 1

Write the given number 120 as the product of its primes. Count the different prime numbers that are appearing in the product to get the total prime factors of 120. If any prime factor is appearing more than once then count it only one time.

Complete answer:
Here, we have been provided with the number 120 and we are asked to find the total number of prime factors it has.
Now, we know that every number can be expressed as the product of its prime factors. We find prime factors of a number to find its square root, cube root generally. But in the above question we just have to count the prime factors of 120. So, we can write 120 as: -
$120=12\times 10$
Now, we need to find the prime factors of 12 and 10 separately and multiply them. So, we have,

& \Rightarrow 12=2\times 2\times 3 \\\ & \Rightarrow 10=2\times 5 \\\ \end{aligned}$$ Therefore, 120 can be written as: - $$\begin{aligned} & \Rightarrow 120=\left( 2\times 2\times 3 \right)\times \left( 2\times 5 \right) \\\ & \Rightarrow 120=\left( 2\times 2\times 2 \right)\times 3\times 5 \\\ \end{aligned}$$ In exponential form, we can write: - $$\Rightarrow 120={{2}^{3}}\times 3\times 5$$ Now, we can see that the prime factor 2 is appearing 3 times but here we have to count it only one time as we have to find the total number of different prime factors. Clearly, we have three different prime factors: they are 2, 3 and 5. So, 120 has three prime factors. **Note:** One may note that if we will count the prime factor ‘2’ three times then the total number of prime factors of 120 would be five, but it will be a wrong answer. You have to understand it yourself that we have to count the identical prime factors only once. You must know how to write a composite number as the product of its primes otherwise you will not be able to solve the question.