Examine each of the following statements, and determine whether it is 'Always true', 'Sometimes true', 'Never true'. Statement 7: If a number is divisible by both 6 and 4, it must be divisible by 24.
Sometimes true. If a number is divisible by both 6 and 4, it is divisible by LCM(6,4) = 12, but not necessarily by 24. For example, 12 is divisible by both 6 and 4, but not by 24. However, 24 is divisible by both 6 and 4, and also by 24.
Explanation
The chapter explains that divisibility by two numbers ensures divisibility by their LCM, not necessarily by their product. LCM(6,4) = 12, not 24. Since 6 = and 4 = 2², the LCM is 2² × 3 = 12. A number divisible by both 6 and 4 must be divisible by 12, but may or may not be divisible by 24. Example: 12 is divisible by 6 and 4, but 12 ÷ 24 is not an integer. Counter-example showing 'sometimes': 48 is divisible by 6, 4, and 24.