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.