shabuzen102 wrote:
cavana wrote:
1) Since n is squared, it should have two identical sets of prime factors
2) Since n^2 is divisible by 96, all prime factors of 96 should be the prime numbers of n^2.
3) The prime factors of 96 are 2*2*2*2*2*3. A squared number that includes all these prime factors should have an additional factor 2 and an additional factor 3.
4) n^2 has to be at least 2^6*3^2, and n has to be 2^3*3=24
5) If n has to be at least 24, then the largest integer it MUST be divisible by is 24.
Posted from my mobile device
Dear
VeritasKarishma,
Would you please explain point 5 - I've followed everything from point 1 to point 4, but it lost me at point 5, which says "if n has to be at least 24, then the largest integer it must be divisible by is 24". The thing is, "n has to be at least 24" doesn't mean that n has to be 24 (nor does it say anywhere in the prompt." If n equaled 48, it still follows everything that the prompt asks. Let's try plugging n - 48 into the question:
"Say n is a positive integer 48, and n^2 is divisible by 96 (to be exact, n^2 = 96 * 24), then the largest positive integer that must divide 48 is 48" => This statement is perfectly fine and there's nothing wrong with n being 48. Does it say anywhere that n has to be 24, or does it only say n has to be at least 24?
I'm very confused by this question and I've looked through Bunuel's similar questions and I understand the process, I just didn't understand why it has to be the least value - what part in the sentence indicates that n has to be least value? Thank you very much. I appreciate your help.
n^2 has 2^5*3 as a factor. Since it is a square, it MUST have 2^6 * 3^2 as factors and can have other factors too such as n^2 can be 2^6 * 3^2 *5^2 or 2^6 * 3^4 or 2^6 * 3^2 * 11^2 etc.
The smallest value of n is then 2^3 * 3 = 24. Of course, if n^2 can take many different values, n can take as many different corresponding values such as 2^3 * 3* 5 or 2^3 * 3^2 or 2^3 * 3* 11 and so on...
What will certainly be a factor of n? 2^3 * 3 (= 24). We don't know about other additional factors.
n can take various values including 24. e.g. 24, 48, 72, 96 ... and so on
Which number MUST divide each of these values of n? 24.
Can I say that 48 MUST divide each of these values of n? No. Because 48 cannot divide 24.
Can I say that 72 MUST divide each of these values of n? No. Because 72 cannot divide 24 and 48.
and so on...