Math Problem Statement
solve
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: The prime divisors of 150 are . So, the sum is: Answer: A) 10
2. Find the number of natural divisors of 300.
We start by factoring 300: To find the number of divisors, we use the formula for the number of divisors of a number , which is given by: For 300, the exponents of the prime factors are 2, 1, and 2. Therefore, the number of divisors is: Answer: B) 18
3. If and , find the sum of all possible values of .
We know that must be a divisor of 36 that is less than 36 and has a gcd of 6 with 36. The divisors of 36 are: Now, we check which of these have gcd 6 with 36. The numbers that satisfy are 6, 12, and 18. The sum of these values is: Answer: A) 36
4. Find and , where .
From the formula , we have: Therefore, the value of is: 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:
- How do you calculate the number of divisors of a number in general?
- What is the role of the gcd in problems involving divisibility and common factors?
- What other methods can be used to find the sum of divisors of a number?
- How do you find the lcm and gcd of two numbers manually without a calculator?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Divisibility, GCD, and Algebraic Factorization Problems
Solve Number Theory Problems on LCM, GCD, and Prime Factorization
Solving Divisibility, GCD, and Modular Arithmetic Problems
Understanding Number Theory Problems: GCD, Prime Numbers, and Functions
Number Theory: Divisors, Floor Functions, and Diophantine Equations