Question 1209946
Let's solve each problem step-by-step.

**1. Finding the 4 Integers**

* Let the 4 integers be $a, b, c, d$.
* We are given:
    * $a + b + c + d = 24$
    * $a \cdot b \cdot c \cdot d = 945$
* First, find the prime factorization of 945:
    * $945 = 3^3 \cdot 5 \cdot 7 = 3 \cdot 3 \cdot 3 \cdot 5 \cdot 7$
* We need to find 4 integers that multiply to 945 and add to 24.
* Let's try different combinations:
    * If we take 3, 5, 7, the remaining factor is 9. 3+5+7+9 = 24. 3*5*7*9 = 945.
* Thus, the integers are 3, 5, 7, and 9.

**2. Sum of Natural Numbers Divisible by 13**

* We need to find the sum of natural numbers between 500 and 1000 that are divisible by 13.
* The first number divisible by 13 greater than 500 is:
    * $500 / 13 \approx 38.46$, so the first number is $39 \cdot 13 = 507$.
* The last number divisible by 13 less than 1000 is:
    * $1000 / 13 \approx 76.92$, so the last number is $76 \cdot 13 = 988$.
* We have an arithmetic progression (AP) with:
    * First term ($a_1$) = 507
    * Common difference ($d$) = 13
    * Last term ($a_n$) = 988
* To find the number of terms ($n$):
    * $a_n = a_1 + (n - 1)d$
    * $988 = 507 + (n - 1)13$
    * $481 = (n - 1)13$
    * $n - 1 = 481 / 13 = 37$
    * $n = 38$
* To find the sum of the AP ($S_n$):
    * $S_n = \frac{n}{2}(a_1 + a_n)$
    * $S_{38} = \frac{38}{2}(507 + 988)$
    * $S_{38} = 19(1495) = 28405$

**3. Consecutive Numbers in AP**

* Let the three consecutive numbers in AP be $a - d$, $a$, and $a + d$.
* We are given:
    * $(a - d) + a + (a + d) = 15$
    * $(a - d)^2 + (a + d)^2 = 58$
* From the first equation:
    * $3a = 15$
    * $a = 5$
* Substitute $a = 5$ into the second equation:
    * $(5 - d)^2 + (5 + d)^2 = 58$
    * $25 - 10d + d^2 + 25 + 10d + d^2 = 58$
    * $50 + 2d^2 = 58$
    * $2d^2 = 8$
    * $d^2 = 4$
    * $d = \pm 2$
* If $d = 2$, the numbers are $5 - 2$, $5$, $5 + 2$, which are 3, 5, 7.
* If $d = -2$, the numbers are $5 - (-2)$, $5$, $5 + (-2)$, which are 7, 5, 3.

**Summary**

1.  The integers are 3, 5, 7, and 9.
2.  The sum is 28405.
3.  The numbers are 3, 5, 7.