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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]  25 Apr 2013, 21:39
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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
[Reveal] Spoiler: OA
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Re: If p.q and r are positive integers greater than 1, and p and [#permalink]  25 Apr 2013, 22:35
Expert's post
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]  26 Apr 2013, 00:29
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]  26 Apr 2013, 00:34
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]  25 Dec 2013, 15:29
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.

Cheers
J
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Re: If p.q and r are positive integers greater than 1, and p and [#permalink]  26 Dec 2013, 00:40
1
KUDOS
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.

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.

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Re: If p.q and r are positive integers greater than 1, and p and   [#permalink] 26 Dec 2013, 00:40
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