Find the number of prime factor of 20790

Options

A. 7
B. 9
C. 11
D. 13

Answer

To find the number of prime factors of 20790, we will perform the prime factorization step by step.

### Step 1: Factor out the obvious factors
Since 20790 ends with a 0, it is divisible by 10. We can factor out 10 first.
\[ 20790 \div 10 = 2079 \]
So, we have:
\[ 20790 = 10 \times 2079 \]

### Step 2: Factor 10 into prime factors
The prime factorization of 10 is:
\[ 10 = 2 \times 5 \]
Now we have:
\[ 20790 = 2 \times 5 \times 2079 \]

### Step 3: Factor 2079
Next, we need to factor 2079. We can check for divisibility by smaller prime numbers. The sum of the digits of 2079 (2 + 0 + 7 + 9 = 18) is divisible by 3, so 2079 is divisible by 3.
\[ 2079 \div 3 = 693 \]
Now we have:
\[ 20790 = 2 \times 5 \times 3 \times 693 \]

### Step 4: Factor 693
Next, we factor 693. The sum of the digits of 693 (6 + 9 + 3 = 18) is also divisible by 3, so 693 is divisible by 3.
\[ 693 \div 3 = 231 \]
Now we have:
\[ 20790 = 2 \times 5 \times 3^2 \times 231 \]

### Step 5: Factor 231
Next, we factor 231. The sum of the digits of 231 (2 + 3 + 1 = 6) is divisible by 3, so 231 is divisible by 3.
\[ 231 \div 3 = 77 \]
Now we have:
\[ 20790 = 2 \times 5 \times 3^3 \times 77 \]

### Step 6: Factor 77
Next, we factor 77. It can be factored into:
\[ 77 = 7 \times 11 \]
Now we have:
\[ 20790 = 2 \times 5 \times 3^3 \times 7 \times 11 \]

### Step 7: Count the prime factors
The prime factorization of 20790 is:
\[ 20790 = 2^1 \times 3^3 \times 5^1 \times 7^1 \times 11^1 \]
To find the number of distinct prime factors, we count the unique prime numbers in the factorization:
- 2
- 3
- 5
- 7
- 11

Thus, the number of distinct prime factors of 20790 is 5.

### Final Answer
The number of prime factors of 20790 is **5**.

---