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If p, q and r are positive integers greater than 1, and p [#permalink]
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25 Apr 2013, 22:39
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If p, q and r are positive integers greater than 1, and p and q are factors of r, which of the following must be the factor of r^(pq)? I. p+q II. q^p III. p^2 * q^2 A. I only B. II only C. III only D. I and II E. II and III
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Re: If p.q and r are positive integers greater than 1, and p and [#permalink]
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25 Apr 2013, 23:35
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rajatr wrote: If p.q and r are positive integers greater than 1, and p and q are factors of r, which of the following must be the factor of r^pq?
I. p+q II. q^p III. p^2 * q^2
A. I only
B. II only
C. III only
D. I and II
E. II and III We know that p and q are factors of r. Thus, for 2 nonnegative integers,m and n, we have : r = pm and r = qn. Now, \(r^{pq} = (pm)^{pq} = (qn)^{pq}\). Thus, we see that \(q^p\) will always be a factor of the given expression. Also, from the given problem p,q >2. Thus, the minimum value of p*q = 4. Thus,\(r^{pq}\) = Atleast \(r^4\). Now, as r = pm = qn, thus, \(r^4 = (p*q*m*n)^2 = p^2*q^2*m^2*n^2.\)Thus, \(p^2 * q^2\) is also a factor of the given expression. E.
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Re: If p, q and r are positive integers greater than 1, and p [#permalink]
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26 Apr 2013, 01:29
given that p and q are factors of r. so we can picture it this way r=p*q*n (nsome another factor of r) so, r^pq= (p*q*n)^pq I. p+q .since the question is MUST BE TRUE, we eleminate this option II. (p*q*n)^pq / q^p= integer YES! III.(p*q*n)^pq/ p^2 * q^2 YEs, since we are said that integer p>1 and integer q>1
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Re: If p, q and r are positive integers greater than 1, and p [#permalink]
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26 Apr 2013, 01:34
LalaB wrote: given that p and q are factors of r. so we can picture it this way r=p*q*n (nsome another factor of r) so, r^pq= (p*q*n)^pq
Hi LalaB, this part is incorrect. Consider the case \(r=30\) and its factor \(p=15\) \(q=10\) According to your statement \(30=10*15*other factor\) As you can see this is not true: no factor of 30 will fit into that equation . (What you say can be true only in some cases....) Hope this clarifies, let me know
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Re: If p.q and r are positive integers greater than 1, and p and [#permalink]
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25 Dec 2013, 16:29
mau5 wrote: rajatr wrote: If p.q and r are positive integers greater than 1, and p and q are factors of r, which of the following must be the factor of r^pq?
I. p+q II. q^p III. p^2 * q^2
A. I only
B. II only
C. III only
D. I and II
E. II and III We know that p and q are factors of r. Thus, for 2 nonnegative integers,m and n, we have : r = pm and r = qn. Now, \(r^{pq} = (pm)^{pq} = (qn)^{pq}\). Thus, we see that \(q^p\) will always be a factor of the given expression. Also, from the given problem p,q >2. Thus, the minimum value of p*q = 4. Thus,\(r^{pq}\) = Atleast \(r^4\). Now, as r = pm = qn, thus, \(r^4 = (p*q*m*n)^2 = p^2*q^2*m^2*n^2.\)Thus, \(p^2 * q^2\) is also a factor of the given expression. E. What about the first statement? Cheers J



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Re: If p.q and r are positive integers greater than 1, and p and [#permalink]
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26 Dec 2013, 01:40
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jlgdr wrote: mau5 wrote: rajatr wrote: If p.q and r are positive integers greater than 1, and p and q are factors of r, which of the following must be the factor of r^pq?
I. p+q II. q^p III. p^2 * q^2
A. I only
B. II only
C. III only
D. I and II
E. II and III We know that p and q are factors of r. Thus, for 2 nonnegative integers,m and n, we have : r = pm and r = qn. Now, \(r^{pq} = (pm)^{pq} = (qn)^{pq}\). Thus, we see that \(q^p\) will always be a factor of the given expression. Also, from the given problem p,q >2. Thus, the minimum value of p*q = 4. Thus,\(r^{pq}\) = Atleast \(r^4\). Now, as r = pm = qn, thus, \(r^4 = (p*q*m*n)^2 = p^2*q^2*m^2*n^2.\)Thus, \(p^2 * q^2\) is also a factor of the given expression. E. What about the first statement? Cheers J For this kind of question, I like to plug in numbers. p = 2 q = 3 r = 6 ==>6^(2*3) = \((6^2)^3 = 36^3\) Clearly, 36 is not divisible by 5 (2+3 =5) Only number ending with 5 or 0 can be divisible by 5 ==> 1st statement is not a "must be true" answer. Best!
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