https://file.aiquickdraw.com/mathbot/file/1726217846406q4r4fcyb.jpg

Let's analyze the questions from the image and solve them step by step.

### 1. **First Question:**
   - **Problem:**
     *Ikkita o'zaro tub sonlarning EKUKi 108 ga teng bo'lsa, bu sonlarning ko'paytmasini toping.*
     (If the LCM of two co-prime numbers is 108, find their product.)

   **Solution:**
   If two numbers are co-prime, their Least Common Multiple (LCM or EKUK) is simply the product of the two numbers. Therefore, if their LCM is 108, their product must also be 108.

   **Answer:**
   \( \boxed{B) 108} \)

### 2. **Second Question:**
   - **Problem:**
     *45 sonining tub bo'luvchilari nechta?*
     (How many prime divisors does 45 have?)

   **Solution:**
   First, find the prime factorization of 45:
   \[
   45 = 3^2 \times 5
   \]
   The prime divisors of 45 are 3 and 5, so there are 2 distinct prime divisors.

   **Answer:**
   \( \boxed{B) 2} \)

### 3. **Third Question:**
   - **Problem:**
     \[
     Hisoblang: \frac{1^3 + 2^3 + 3^3 + 4^3}{(2 + 3)^2}
     \]
     (Calculate the given expression.)

   **Solution:**
   Let's break this down:
   \[
   1^3 + 2^3 + 3^3 + 4^3 = 1 + 8 + 27 + 64 = 100
   \]
   \[
   (2 + 3)^2 = 5^2 = 25
   \]
   So, the expression becomes:
   \[
   \frac{100}{25} = 4
   \]

   **Answer:**
   \( \boxed{C) 4} \)

### 4. **Fourth Question:**
   - **Problem:**
     Given \( a = 2^3 \times 3^2 \times 5 \) and \( b = 2^2 \times 3^3 \times 5^2 \), find \( \frac{EKUK(a, b)}{EKUB(a, b)} \).
   
   **Solution:**
   - EKUK (LCM) is found by taking the highest powers of all prime factors:
     \[
     EKUK(a, b) = 2^3 \times 3^3 \times 5^2
     \]
   - EKUB (GCD) is found by taking the lowest powers of all prime factors:
     \[
     EKUB(a, b) = 2^2 \times 3^2 \times 5
     \]
   Now, calculate:
   \[
   \frac{EKUK(a, b)}{EKUB(a, b)} = \frac{2^3 \times 3^3 \times 5^2}{2^2 \times 3^2 \times 5} = 2^{3-2} \times 3^{3-2} \times 5^{2-1} = 2 \times 3 \times 5 = 30
   \]

   **Answer:**
   \( \boxed{B) 30} \)

Would you like more details on any of these solutions, or do you have other questions?

Here are 5 related questions to practice:
1. If two numbers are co-prime and their product is 72, what is their LCM?
2. What are the prime divisors of 60?
3. Calculate the sum of cubes from \(1^3\) to \(5^3\).
4. Given \( a = 2^4 \times 3 \times 7 \) and \( b = 2^2 \times 3^3 \times 7^2 \), find \( \frac{EKUK(a, b)}{EKUB(a, b)} \).
5. If two numbers have an EKUB (GCD) of 12 and an EKUK (LCM) of 360, what is their product?

**Tip:** Always factorize numbers into primes when solving LCM or GCD problems.

Ikkita o'zaro tub sonlarning EKUKi 108 ga teng bo'lsa, bu sonlarning ko'paytmasini toping. 45 sonining tub bo'luvchilari nechta? Hisoblang: 1^3 + 2^3 + 3^3 + 4^3 / (2 + 3)^2. a = 2^3 . 3^2 . 5; b = 2^2 . 3^3 . 5^2 bo'lsa, EKUK(a,b)/EKUB(a,b) ning qiymatini toping.

Learn how to solve number theory problems, including finding LCM and GCD of co-prime numbers, counting prime divisors, and simplifying expressions involving cubes and powers. This lesson is ideal for students in Grades 9-11.

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