Find total number of distinct primes in the prime factorization of number 27300

Apart from this post. We have already solved many of the questions related to prime factorization. But here in this post we will not only find the prime factor of 27300 but also know what is distinct prime?

So before we start calculation of prime factors of 27300 l. Let’s understand the concept of distinct prime.

As we know prime factors is the multiple of that number which we have to extract prime factors. In more simple world a prime factor is the multiple of the given number. Even each factors are the multiple of given number. 

Suppose we have a given number ‘X’. Now let’s suppose the prime factor of number ‘X’ is 1,2,3,4,5, and 6. 

Now, the distinct prime is the collection of all the numbers which is a multiple of given number but each are different-different value.

Means a distinct prime factor is a collection of those numbers which are not repeating.

First of all let’s find the prime factor of 27300.

2 |27300 

2 |13650

3 |6825

5 |2275

5 |455

5| 91

Hence, the prime factor of 27300 is 2 × 2 × 3 × 5 × 5 × 5 × 91

Therefore, the distinct prime of 27300 will be 2 × 2 × 3 × 5 × 5 × 5 × 91.

To find the total number of distinct primes in the prime factorization of the number 27300, we begin by factoring 27300.

Step 1: Divide by 2 (smallest prime)

27300 is an even number, so divide by 2:

$$27300 div 2 = 13650$$

13650 is still even, so divide by 2 again:

$$13650 div 2 = 6825$$

Now, 6825 is odd, so we stop dividing by 2.

Step 2: Divide by 3 (next smallest prime)

Check if 6825 is divisible by 3 (sum of digits = 6 + 8 + 2 + 5 = 21, which is divisible by 3):

$$6825 div 3 = 2275$$

Now, 2275 is not divisible by 3.

Step 3: Divide by 5 (next smallest prime)

Since 2275 ends in 5, it’s divisible by 5:

$$2275 div 5 = 455$$

455 also ends in 5, so divide by 5 again:

$$455 div 5 = 91$$

Now, 91 is not divisible by 5.

Step 4: Divide by 7 (next smallest prime)

Check if 91 is divisible by 7:

$$91 div 7 = 13$$

Now we have 13, which is a prime number.

Step 5: Prime factorization of 27300

We now have the complete prime factorization of 27300:

$$27300 = 2^2 times 3 times 5^2 times 7 times 13$$

Step 6: Count the distinct primes

The distinct primes in the factorization are: 2, 3, 5, 7, and 13.

Thus, the total number of distinct primes is 5.