Math Problem Statement

Determine whether each statement below is True or False. Explain briefly if the statement is true or give a counterexample if it is false. (a) If 2 and 3 are factors of a number, then 12 must be a factor of that number. (b) If 2 or 5 are factors of a number, then 10 must be a factor of that number. (c) If 15 is a multiple of a number, then 45 must be a multiple of that number.

Solution

Let's analyze and solve each of the statements step-by-step.

(a) If 2 and 3 are factors of a number, then 12 must be a factor of that number.

  • Explanation:
    If 2 and 3 are factors of a number, their least common multiple (LCM) is 6. This means any number divisible by both 2 and 3 is divisible by 6. However, this does not guarantee that the number is divisible by 12 unless 4 (a factor of 12 but not of 6) is also a factor.

  • Counterexample:
    Consider the number 6. Both 2 and 3 are factors of 6, but 12 is not a factor of 6.

  • Answer: False.

(b) If 2 or 5 are factors of a number, then 10 must be a factor of that number.

  • Explanation:
    If a number has 2 or 5 as factors, this does not imply that 10 (the product of 2 and 5) is a factor. For 10 to be a factor, both 2 and 5 must be factors simultaneously.

  • Counterexample:
    Consider the number 4. It has 2 as a factor but not 5, so 10 is not a factor of 4. Similarly, consider the number 5, which has 5 as a factor but not 2.

  • Answer: False.

(c) If 15 is a multiple of a number, then 45 must be a multiple of that number.

  • Explanation:
    If 15 is a multiple of a number, it implies the number is a factor of 15. For 45 to also be a multiple of that number, the number must divide both 15 and 45. This is true because 45 is also a multiple of 15. Any factor of 15 will also be a factor of 45.

  • Answer: True.

Final Results:

  • (a) False
  • (b) False
  • (c) True

Would you like me to explain any part in greater detail or clarify?

Here are 5 related questions to expand your understanding:

  1. What is the least common multiple (LCM) of two numbers, and how is it calculated?
  2. How do we determine if a number is a factor of another number?
  3. Can the product of two numbers always be a factor of their LCM? Why or why not?
  4. What is the difference between "factor" and "multiple"?
  5. How can counterexamples be effectively used to disprove a mathematical statement?

Tip: To verify whether a statement is true or false, try using small numbers as examples or counterexamples to test the logic.

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Math Problem Analysis

Mathematical Concepts

Factors and multiples
Least common multiple (LCM)
Counterexamples in mathematics

Formulas

LCM(a, b) = smallest number divisible by both a and b

Theorems

If a number is divisible by two other numbers, it is divisible by their LCM.

Suitable Grade Level

Grades 6-8

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