If k and x are positive integers and x is divisible by 6, which of the following CANNOT be the value of ?
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Wait, I'm confused. How can 24√k be an option? If k=1, then 24√1 = 24. But the expression is √(288kx) which simplifies to 24√(2k) if x=6. That's not 24√k. So B cannot be the value?
Because x is divisible by 6, so x=6n. Then √(288k*6n) = √(1728kn) = 24√(3kn). Since n is a positive integer, not necessarily 1. So you need to see if 24√k can be expressed as 24√(3kn) for some n. That would require 3kn = k, so n=1/3, not integer. So B is impossible. But wait, the question asks which CANNOT be, and we need to check all options.
Actually, 24√k is possible if k is a multiple of something? Let me think. If 3kn = k, then n=1/3, not integer. So B is impossible. But let's check others.
I got C as the answer. Let's see: √(288kx) = √(288k*6n) = √(1728kn) = 24√(3kn). For C to be 24√(3k), we need 3kn = 3k => n=1. That's possible since n=1 is positive integer (x=6). So C is possible. For A: 24k√3 = 24√(3k^2) => need 3kn = 3k^2 => n=k. That's possible. For D: 24√(6k) => need 3kn = 6k => n=2, possible. For E: 72√k = 24√(9k) => need 3kn = 9k => n=3, possible. For B: need 3kn = k => n=1/3, impossible. So B is the answer. But the user above said B. I agree with B. But the question says 'which CANNOT be the value' and B is the one. So answer is B.
But wait, the choices are (A) 24k√3 (B) 24√k (C) 24√(3k) (D) 24√(6k) (E) 72√k. So B is 24√k, which cannot be. But I've seen some say B is possible if k=1/9? But k is positive integer, so no.
This is a bit tricky because you have to consider that x can be any multiple of 6. I initially thought B was possible because I set x=6, then got 24√(3k) which is C. So B is not possible. But then I realized that D and E are also possible by choosing x appropriately. So B is the only one that can't. Nice question.
Can someone explain why A is possible? I get that we need 3kn = 3k^2, so n=k. But then x=6k. That's fine since k is positive integer. So A is possible. But then why is B not possible? Because n would have to be 1/3. So B is the answer. Thanks!
What it tests
Your command of divisibility, primes, factors, parity, and how the GMAT tests properties of integers.
Common trap
Forgetting that 1 is not prime, that 0 is divisible by every integer, or that negative numbers flip your assumptions.