Final answer:
The LCM of 8, 6, 8n, 6nm, and 4,2,4n² is 24 since this number contains the highest powers of prime factors from all the given numbers, considering n and m are prime numbers. The correct option is (b) 24
Step-by-step explanation:
The student's question asks for the least common multiple (LCM) of a given set of numbers: 8, 6, 8n, 6nm, and 4,2,4n², where n and m are prime numbers. To solve the mathematical problem completely, we need to find the LCM of these numbers.
Firstly, we can simplify the problem by noting that the LCM of 8, 6, and 4 is 24. Since n and m are prime numbers, the terms 8n and 4n² would also be multiples of 8 and 4, respectively, hence included in the LCM of 24 already.
For 6nm, as long as m is a different prime number from 6, the LCM must include this prime factor. However, since 6 is composed of the prime factors 2 and 3, and because we already have multiples of 8 (2³) in our LCM, the additional factor from 6 that we need is just 3.
Thus, the LCM of these numbers is 24 (which is 8 * 3), so the correct option is (b) 24. This is because 24 contains the highest powers of prime factors from all the given numbers.