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Math Expert V
Joined: 02 Sep 2009
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If r and s are positive integers, is the greatest prime factor of r la  [#permalink]

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If r and s are positive integers, is the greatest prime factor of r larger than the greatest prime factor of s?

(1) r = 11^s
(2) r/s>1

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Current Student D
Joined: 12 Aug 2015
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GRE 1: Q169 V154 Re: If r and s are positive integers, is the greatest prime factor of r la  [#permalink]

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Here is my solution
From the first statement => if s=2 ; r = 11^2 => the answer is yes ; but if s=11 and r=11^11 the answer is no
Not sufficient

from two => no info on the values => not sufficient.
We can use the same example as in A to say that E is the answer

My answer is E
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Re: If r and s are positive integers, is the greatest prime factor of r la  [#permalink]

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looks like E.
so the Qs presupposes that r and s MUST have prime factors?
cos if you if u plug in s=1, then s has no prime factor to start off with.
Am I terribly wrong?

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Intern  G
Joined: 18 Nov 2013
Posts: 46
Re: If r and s are positive integers, is the greatest prime factor of r la  [#permalink]

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Its E.

First statement r=11^s

r=11 and s=1
r=11and s=11; Hence 2 options; Not suff

Second statement: (r/s)>1

r can 10 and s can be 5
r can 11 and s can be 2

Combining both;
r can be 11^x and s can be 11- Highest prime factor for both r and s is 11
r can be 11^x and s can be 2- Highest prime factor for r is 11 and s is 2

Hence E.
Current Student D
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GRE 1: Q169 V154 If r and s are positive integers, is the greatest prime factor of r la  [#permalink]

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Hi chetan2u
I have a query here.
How many prime factors does 1 have ?
Zero right ?

Regards
Stone Cold

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Re: If r and s are positive integers, is the greatest prime factor of r la  [#permalink]

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stonecold wrote:
Hi chetan2u
I have a query here.
How many prime factors does 1 have ?
Zero right ?

Regards
Stone Cold

Hi,

1 is not a prime number.
A prime number has Only two POSITIVE factors :- 1 and itself.
And a composite number has more than TWO positive factors.

However 1 has only 1 POSITIVE factor, so it is neither prime nor composite. It is a unity or a natural number
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If r and s are positive integers, is the greatest prime factor of r la  [#permalink]

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Hi niks18 VeritasKarishma chetan2u Bunuel PKN

Is the question based on prime factorization of positive integers?

If r and s are positive integers, is the greatest prime factor of r larger than the greatest prime factor of s?

Quote:
(1) r = 11^s

I initially thought that any prime no raised to any positive exponent will have same no of prime factors
as the no itself.
e.g. 11^s where s is positive integer will have same no of prime factors as 11.
So suff. prime factors of r and s are same: 1 and 11

But statements also opens up the possibility of s being a composite no.
E.g. 11^38; 38 which is 19*2
Now prime factors of s: 19, 2 and r: 11,19,2; r has more prime factors than s

So St 1 is insuff

Quote:
(2) r/s>1

r>s
r=4, s =2 , prime factors of r and s are equal
r=6, s =2 , prime factors of r(2,3) are more than s (2)
St Insuff

Combining St 1 and St 2
Now, this got trickier.
A prime no raised to any positive integer does not add any value to fact:
the prime no raised to exponent is greater than the prime no itself.
What is the use of St 2 while combining with St 1?
How do I link this to prime factorization of the number itself?
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Re: If r and s are positive integers, is the greatest prime factor of r la  [#permalink]

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1
Bunuel wrote:
If r and s are positive integers, is the greatest prime factor of r larger than the greatest prime factor of s?

(1) r = 11^s
(2) r/s>1

Note that greatest prime factor of a large number could be very small. e.g. 2^100 is a very large number but its greatest prime factor is 2 only whereas the greatest prime factor of 26, a relatively small number, is 13.

(1) r = 11^s
This tells us that the greatest prime factor of r is 11. We don't know the greatest prime factor of s. s could be 2 or 17 or 101 etc

(2) r/s>1
This means r > s.
As we saw above, greatest prime factor of a large number could be very small. So higher value is no guarantee of a large prime factor.

Using both,
r could be 11^2 (greatest prime factor 11) in which case s = 2 (greatest prime factor 2) or
r could be 11^13 (greatest prime factor 11) in which case s = 13 (greatest prime factor 13)
Both cases are possible. So not sufficient.
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