Examine each of the following statements, and determine whether it is 'Always true', 'Sometimes true', 'Never true'. Statement 6: If a number is divisible by both 9 and 4, it must be divisible by 36.
Always true. If a number is divisible by both 9 and 4, it must be divisible by LCM(9,4) = 36. Since 9 = 3² and 4 = 2² are coprime (no common factors), their LCM is = 36.
Explanation
The chapter explains that if a number is divisible by two numbers, it is divisible by their LCM. For 9 and 4: 9 = 3² and 4 = 2². Since 9 and 4 have no common prime factors (they are coprime), LCM(9,4) = = 36. Any number divisible by both 9 and 4 must contain both 3² and 2² in its prime factorization, making it divisible by 36. Example: 72 is divisible by 9 (72 = ) and by 4 (72 = ), and 72 = .