Math Problem Statement

Which statement is true? Group of answer choices

The least common multiple of 4 and 8 is 16.

The least common multiple of 6 and 10 is 30

The least common multiple of 4 and 10 is 40.

The least common multiple of 6 and 8 is 48.

Solution

Let's evaluate the least common multiples (LCM) for each pair of numbers:

  1. LCM of 4 and 8: The factors of 4 are 4=224 = 2^2 and the factors of 8 are 8=238 = 2^3. The LCM takes the highest power of all prime factors, so the LCM is 23=82^3 = 8.
    This statement is false (LCM is 8, not 16).

  2. LCM of 6 and 10: The factors of 6 are 6=2×36 = 2 \times 3 and the factors of 10 are 10=2×510 = 2 \times 5. The LCM is found by taking the highest power of each prime factor:
    LCM(6,10)=2×3×5=30LCM(6, 10) = 2 \times 3 \times 5 = 30.
    This statement is true.

  3. LCM of 4 and 10: The factors of 4 are 4=224 = 2^2 and the factors of 10 are 10=2×510 = 2 \times 5. The LCM is 22×5=202^2 \times 5 = 20.
    This statement is false (LCM is 20, not 40).

  4. LCM of 6 and 8: The factors of 6 are 6=2×36 = 2 \times 3 and the factors of 8 are 8=238 = 2^3. The LCM is 23×3=242^3 \times 3 = 24.
    This statement is false (LCM is 24, not 48).

Conclusion:

The true statement is:
The least common multiple of 6 and 10 is 30.

Would you like any further details or explanations?

Here are 5 relative questions to expand this concept:

  1. How do you find the greatest common divisor (GCD) of two numbers?
  2. What is the relationship between GCD and LCM for two numbers?
  3. How do you calculate the LCM of three numbers?
  4. What are prime factorization and its role in finding the LCM?
  5. Can the LCM of two numbers ever be smaller than one of the numbers?

Tip: The LCM of two numbers is always greater than or equal to the larger number.

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Math Problem Analysis

Mathematical Concepts

Least Common Multiple (LCM)
Prime Factorization
Multiples

Formulas

LCM formula: Use the highest powers of all prime factors

Theorems

Prime Factorization Theorem

Suitable Grade Level

Grades 6-8