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If p, q and r are positive integers greater than 1, and p

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If p, q and r are positive integers greater than 1, and p [#permalink]

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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
[Reveal] Spoiler: OA

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Re: If p.q and r are positive integers greater than 1, and p and [#permalink]

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New post 25 Apr 2013, 23:35
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 non-negative 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,\(r^{pq}\) = At-least \(r^4\). 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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New post 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 (n-some 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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New post 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 (n-some 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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New post 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 non-negative 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,\(r^{pq}\) = At-least \(r^4\). 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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New post 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 non-negative 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,\(r^{pq}\) = At-least \(r^4\). 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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Re: If p, q and r are positive integers greater than 1, and p [#permalink]

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Re: If p, q and r are positive integers greater than 1, and p   [#permalink] 13 Oct 2017, 08:00
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