The number N is the product of two prime numbers and has a positive positive sum smaller than N with a value of 2020. Find N value

To solve the problem, let \( N \) be the product of two prime numbers, say \( p \) and \( q \), so:

\[
N = p \times q
\]

We are also given that the sum of the divisors of \( N \), denoted by \( \sigma(N) \), is smaller than \( N \) and equals 2020. The formula for the sum of divisors of \( N \) when \( N = p \times q \) (with \( p \) and \( q \) being distinct prime numbers) is:

\[
\sigma(N) = (1 + p)(1 + q)
\]

We are told that:

\[
\sigma(N) = 2020
\]

Thus, we have the equation:

\[
(1 + p)(1 + q) = 2020
\]

To find the prime factors \( p \) and \( q \), we need to factor 2020 and check for pairs of factors that satisfy this equation.

Factorizing 2020:

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

The divisors of 2020 are: 1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010, 2020.

We are looking for two numbers \( 1 + p \) and \( 1 + q \) among these divisors that are primes \( p \) and \( q \). The prime numbers among the possible values of \( 1 + p \) and \( 1 + q \) are:

\[
2, 5, 101
\]

Thus, the corresponding \( p \) and \( q \) values are \( p = 101 \) and \( q = 5 \). Therefore, the value of \( N \) is:

\[
N = p \times q = 101 \times 5 = 505
\]

Finally, the value of \( N \) is:

\[
N = 505
\]

Would you like further details or explanations?

Here are five related questions you might find helpful:
1. How is the sum of divisors formula derived for any number?
2. What are the properties of prime numbers used in this solution?
3. How do you factor large numbers efficiently?
4. Why is the sum of divisors of a prime product smaller than the product itself?
5. What other methods can be used to find prime factors of a given number?

**Tip:** When solving divisor-related problems, it's helpful to memorize the sum of divisors formula for different types of numbers (e.g., prime, prime powers, and products of primes).

The number N is the product of two prime numbers and has a positive sum smaller than N with a value of 2020. Find N value.

Solve a number theory problem involving the product of two prime numbers where the sum of divisors is smaller than N and equals 2020. The detailed solution includes key concepts in prime numbers and divisor theory, suitable for students in Grades 10-12.

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