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13 Sep 2022, 07:38
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
58% (01:56) correct
42%
(02:00)
wrong
based on 385
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If p is a positive integer and p^3 is divisible by 576, the largest positive integer that must divide P is...
(A) 2
(B) 4
(C) 6
(D) 12
(E) 24
(A) 2
(B) 4
(C) 6
(D) 12
(E) 24
The correct answer here is D) 12.
I have a doubt that if I consider P = 24 then P^3 will be 24*24*24 then it will be divisible by 576 and the greatest integer which will divide P will be 24, how am I wrong?
I have a doubt that if I consider P = 24 then P^3 will be 24*24*24 then it will be divisible by 576 and the greatest integer which will divide P will be 24, how am I wrong?
13 Sep 2022, 08:15
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yashrajrathi20
Notice this: since p is an integer then prime factorization of p^3 will be something like this:
- \(p^3 = a^{(some \ multiple \ of \ 3)}*b^{(some \ multiple \ of \ 3)}*c^{(some \ multiple \ of \ 3)}*...\), where a, b, and c are primes of p.
For example, p^3 could be:
- \(p^3=2^9*3^6*11^{333}\). (Notice that the powers of primes are all multiples of 3.)
While, p^3 cannot be for example:
- \( p^3=2^4*5^6\). In this case \(p=50\sqrt{2}\), which is NOT an integer.
Next, we are told that p^3 is divisible by 576;
\(p^3 = 576*k\), for some positive integer k;
\(p^3 = 2^6*3^2*k\);
As discussed above, k must complete the powers of 2 and 3 to some multiple of 3. The power of 2 is already a multiple of 3, so k must complete 3^2 to a multiple of 3. Thus, the LEAST value of k is 3.
So, the LEAST value of p^3 is \(p^3 = 2^6*3^2*3=2^6*3^3\). This in turn gives the LEAST value of p as \(p =\sqrt[3]{2^6*3^3} =2^2*3=12\).
Of course, k could be more that 3. For example, k could be 3^7*17^99 (notice that in this case k also completes primes of p^3 to some multiples of 3) but the LEAST value of k is 3, and thus the least value of p is 12..
Now, the question asks: what is the largest positive integer that MUST divide p?" Since the least value of p is 12, then we can guarantee that p MUST be divisible by 12! p COULD be divisible by infinitely many larger numbers but that's not what the question asks!
Answer: D.
General Discussion
13 Sep 2022, 09:08
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Bunuel thanks for the reply, you mentioned that the least value of P is 24 in your answer. The question is the largest positive integer that must divide P, so shouldn't that be 24? then the answer would be E, right? I think its just a typing mistake haha.
