How many positive integer factors of 2020 have more than 3 factors?

To solve this problem, we need to find the number of positive integer factors of \( 2020 \) that themselves have more than 3 factors. Here's how we can approach the solution step by step.

### Step 1: Prime Factorization of 2020
We start by finding the prime factorization of \( 2020 \).

\[
2020 = 2^2 \times 5 \times 101
\]

### Step 2: Total Number of Factors of 2020
To calculate the total number of factors of a number, we use its prime factorization. If a number \( N \) has the prime factorization:

\[
N = p_1^{e_1} \times p_2^{e_2} \times \cdots \times p_k^{e_k}
\]

Then, the total number of factors of \( N \) is given by:

\[
(e_1 + 1)(e_2 + 1) \cdots (e_k + 1)
\]

For \( 2020 = 2^2 \times 5^1 \times 101^1 \), the total number of factors is:

\[
(2+1)(1+1)(1+1) = 3 \times 2 \times 2 = 12
\]

So, 2020 has 12 factors in total.

### Step 3: Listing All Factors of 2020
The factors of \( 2020 \) are:

\[
1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010, 2020
\]

### Step 4: Identifying Factors with More Than 3 Factors
Now we need to identify which of these factors themselves have more than 3 divisors.

- \( 1 \): has 1 factor (itself).
- \( 2 \): has 2 factors (\(1, 2\)).
- \( 4 \): has 3 factors (\(1, 2, 4\)).
- \( 5 \): has 2 factors (\(1, 5\)).
- \( 10 \): has 4 factors (\(1, 2, 5, 10\)) → More than 3 factors.
- \( 20 \): has 6 factors (\(1, 2, 4, 5, 10, 20\)) → More than 3 factors.
- \( 101 \): has 2 factors (\(1, 101\)).
- \( 202 \): has 4 factors (\(1, 2, 101, 202\)) → More than 3 factors.
- \( 404 \): has 6 factors (\(1, 2, 4, 101, 202, 404\)) → More than 3 factors.
- \( 505 \): has 4 factors (\(1, 5, 101, 505\)) → More than 3 factors.
- \( 1010 \): has 8 factors (\(1, 2, 5, 10, 101, 202, 505, 1010\)) → More than 3 factors.
- \( 2020 \): has 12 factors → More than 3 factors.

### Step 5: Conclusion
The factors of \( 2020 \) that have more than 3 factors are:

\[
10, 20, 202, 404, 505, 1010, 2020
\]

There are **7 factors** of \( 2020 \) that have more than 3 factors.

Would you like further clarification or a more detailed breakdown of any step?

### Related Questions:
1. How do you find the prime factorization of large numbers?
2. What is the general method for determining the number of divisors of a number?
3. How can you determine if a number has exactly 3 factors?
4. What are the properties of numbers with a small number of divisors?
5. How can we use the prime factorization to solve problems about divisibility?

### Tip:
To quickly determine if a factor has more than 3 divisors, use its prime factorization to count its divisors directly.

Discover how to find the number of positive integer factors of 2020 that have more than 3 divisors. This problem explores prime factorization, divisor counting, and the Divisor Counting Theorem. Ideal for students in Grades 8-10.

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