balboa wrote:

If k is a positive integer, is k the square of an integer?

1) k is divisible by 4.

2) k is divisible by exactly four different prime numbers.

Another way to phrase this is: Is \(sqrt{k}\) an integer?

#1) Insufficient. What are numbers divisible by 4? 4, 8, 12, 16, 20. The question is a yes/no question, so in order to answer the question, the answer must be ALWAYS yes or ALWAYS no.

Since k could be any factor of 4, the possibilities are not always going to be a square of an integer, and there are some instances where k will be a square so it's sometimes yes K is a square of an integer (like when k is 4, 16, 36, 64, etc) but the answer can also be No, k is NOT a square of an integer (like when k = 8, 12, 20, 24, etc).

#2) First figure out what this is telling us. What number are divisible by exactly 4 different prime numbers? This could be 2*3*5*7 = 210. This is not a perfect square. It seems like the statement may be sufficient and the answer is No, when k is divisible by exactly 4 different prime numbers, k will never be the square of an integer, but all we have to do is find 1 instance where k does equal a perfect squre and the statement is insufficient.

Lets think of some perfect squres. 4, 9, 16, 25, 36, 49, 64, 81, 100.

4 = nope 2*2 (only 1 prime, not 4 different prime numbers)

9 = 3*3 (only 1 prime number, not 4 different prime numbers)

16 = 2*2*2*2. Ok, here is where we might be able to say yes. Does the term "different" mean that no prime number can be repeated, or just that there will be 4 terms, and terms may be the same as in this instance.

Can someone say what they think the question means?

#2)

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J Allen Morris

**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.

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