Math Problem Statement

solve

Solve the problems involving prime divisors, number of divisors, gcd, and lcm for various numbers.

Solution

Let's go through each of the questions and solve them step by step.

1. Find the sum of the different prime divisors of 150.

We start by factoring 150: 150=2×3×52150 = 2 \times 3 \times 5^2 The prime divisors of 150 are 2,3,52, 3, 5. So, the sum is: 2+3+5=102 + 3 + 5 = 10 Answer: A) 10

2. Find the number of natural divisors of 300.

We start by factoring 300: 300=22×3×52300 = 2^2 \times 3 \times 5^2 To find the number of divisors, we use the formula for the number of divisors of a number n=p1e1×p2e2××pkekn = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k}, which is given by: (e1+1)(e2+1)(ek+1)(e_1 + 1)(e_2 + 1) \ldots (e_k + 1) For 300, the exponents of the prime factors are 2, 1, and 2. Therefore, the number of divisors is: (2+1)(1+1)(2+1)=3×2×3=18(2+1)(1+1)(2+1) = 3 \times 2 \times 3 = 18 Answer: B) 18

3. If a<36a < 36 and gcd(a,36)=6\text{gcd}(a, 36) = 6, find the sum of all possible values of aa.

We know that aa must be a divisor of 36 that is less than 36 and has a gcd of 6 with 36. The divisors of 36 are: 1,2,3,4,6,9,12,18,361, 2, 3, 4, 6, 9, 12, 18, 36 Now, we check which of these have gcd 6 with 36. The numbers that satisfy gcd(a,36)=6\text{gcd}(a, 36) = 6 are 6, 12, and 18. The sum of these values is: 6+12+18=366 + 12 + 18 = 36 Answer: A) 36

4. Find lcm(a,b)=640\text{lcm}(a, b) = 640 and gcd(a,b)=18\text{gcd}(a, b) = 18, where a×b=lcm(a,b)×gcd(a,b)a \times b = \text{lcm}(a, b) \times \text{gcd}(a, b).

From the formula a×b=lcm(a,b)×gcd(a,b)a \times b = \text{lcm}(a, b) \times \text{gcd}(a, b), we have: a×b=640×18=11520a \times b = 640 \times 18 = 11520 Therefore, the value of a×ba \times b is: 1152011520 Answer: A) 11520


Would you like further details on these solutions or have any questions about the steps? Here are some related questions to expand your learning:

  1. How do you calculate the number of divisors of a number in general?
  2. What is the role of the gcd in problems involving divisibility and common factors?
  3. What other methods can be used to find the sum of divisors of a number?
  4. How do you find the lcm and gcd of two numbers manually without a calculator?
  5. What are the differences between the lcm and gcd, and when would you use each in problem-solving?

Tip: When calculating the number of divisors, always begin with prime factorization to break down the problem step-by-step.

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Math Problem Analysis

Mathematical Concepts

Prime Factorization
Divisors
GCD (Greatest Common Divisor)
LCM (Least Common Multiple)

Formulas

Number of divisors formula: (e1+1)(e2+1)...(ek+1) for n = p1^e1 * p2^e2 * ... * pk^ek
GCD and LCM relationship: a * b = GCD(a, b) * LCM(a, b)

Theorems

Prime Factorization Theorem
GCD and LCM Properties

Suitable Grade Level

Grades 7-9