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605-655 Level|   Number Properties|                     
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Is one not a divisor?
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Bunuel
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Transcendentalist
Is one not a divisor?

It is but its' not even (how many different positive even divisors does n have, including n).
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p can get any prime number greater than 2 so there should be unlimited different divisor for the n because p can be 11,13,17,19....

What is wrong with this ?
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Bunuel
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If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ?

(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight

p can get any prime number greater than 2 so there should be unlimited different divisor for the n because p can be 11,13,17,19....

What is wrong with this ?

No matter which prime p is, 4p will have only four EVEN divisors: 2, 4, 2p, and 4p. Try to check it with any prime greater than 2.

Does this make sense?
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lool
p can get any prime number greater than 2 so there should be unlimited different divisor for the n because p can be 11,13,17,19....

What is wrong with this ?


4 has two even divisors >> 2 & 4

Any prime no p divisors will have 2p & 4p (from above)

So total = 4
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n=4*p

And p(prime) > 2

n= 2*2*p

Positive even divisors 'n' can have including 'n':

2
2p
4
4p which is also equal to 'n'

Hence 4 positive even divisors.
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MensaNumber
n= 4p = 2^2*p^1
so total no. of factors: (2+1)*(1+1)= 6
total no. of odd factors, since p is odd as it is a prime>2: p^1 and p^0 = 2
Total no. of even factors: 6 - 2 = 4

Now if n was n=4pq where p and q are both prime no.s greater than 2 then:
total no. of factors: (2+1)*(1+1)*(1+1)= 12
total no. of odd factors, since p is odd as it is a prime>2: (1+1)*(1+1)= 4
Total no. of even factors: 12 - 4 = 8

Hi Bunuel, could you validate my logic pls?

Yes, your approach is correct.
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ITMRAHUL
If n=4p, where p is a Prime number greater than 2, how many different positive even divisors does n have including n?

a) 2
b) 3
c) 4
d) 6
e) 8

can some 1 throw light on it?

@p=3, n = 4*3 = 12, Positive Even divisor of n = {2, 4, 6, 12} i.e. 4 Divisors
@p=5, n = 4*5 = 20, Positive Even divisor of n = {2, 4, 10, 20} i.e. 4 Divisors
@p=7, n = 4*7 = 28, Positive Even divisor of n = {2, 4, 14, 28} i.e. 4 Divisors
@p=11, n = 4*11 = 44, Positive Even divisor of n = {2, 4, 22, 44} i.e. 4 Divisors
@p=13, n = 4*13 = 52, Positive Even divisor of n = {2, 4, 26, 52} i.e. 4 Divisors
and so on...

Answer: option C
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Hi All,

This question can be solved rather easily by TESTing VALUES:

We're told that N = 4P and that P is a prime number greater than 2. Let's TEST P = 3; so N = 12

The question now asks how many DIFFERENT positive EVEN divisors does 12 have, including 12?

12:
1,12
2,6
3,4

How many of these divisors are EVEN? 2, 4, 6, 12 …..4 even divisors.

Final Answer:
GMAT assassins aren't born, they're made,
Rich
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Hi All,

This question can be solved rather easily by TESTing VALUES:

We're told that N = 4P and that P is a prime number greater than 2. Let's TEST P = 3; so N = 12

The question now asks how many DIFFERENT positive EVEN divisors does 12 have, including 12?

12:
1,12
2,6
3,4

How many of these divisors are EVEN? 2, 4, 6, and 12 …..that's a total of 4 even divisors.

Final Answer:
GMAT assassins aren't born, they're made,
Rich
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n = 4p
p prime no. \(> 2\)

We have to find the no. of even divisors which means even factors of n

P must be a odd no. because 2 is the only even prime no.

Let p be \(3\)

\(n= 4 * 3\)

= \(2^{2} * 3\)

Now , No of factors of P = \(2^{2+1} * 3^{1+1}\) = \(2^{3} * 3^{2}\)

= \(3 * 2 = 6\)

Even factors = Total factors - odd factors

To find Odd factors we take all the prime apart from 2. so here we are left with only 3

Odd factors = \(3^{1+1} = 3^{2}= 2\)

Total factors - Odd factors = Even factors
\(6-2 = 4\)

hence answer is 4
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Walkabout
If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ?

(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight

MATH REVOLUTION VIDEO SOLUTION:

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Walkabout
If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ?

(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight

This is an interesting question because we are immediately given the option to insert any prime number we wish for p. Since this is a problem-solving question, and there can only be one correct answer, we can select any value for p, as long as it is a prime number greater than 2. We always want to work with small numbers, so we should select 3 for p. Thus, we have:

n = 4 x 3

n = 12

Next we have to determine all the factors, or divisors, of P. Remember the term factor is synonymous with the term divisor.

1, 12, 6, 2, 4, 3

From this we see that we have 4 even divisors: 12, 6, 2, and 4.

If you are concerned that trying just one value of p might not substantiate the answer, try another value for p. Let’s say p = 5, so

n = 4 x 5

n = 20

The divisors of 20 are: 1, 20, 2, 10, 4, 5. Of these, 4 are even: 20, 2, 10 and 4. As we can see, again we have 4 even divisors.

No matter what the value of p, as long as it is a prime number greater than 2, n will always have 4 even divisors.

The answer is C.
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Walkabout
If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ?

(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight

We can solve a lot of Integer Properties questions by testing a value

If p is prime, let's let p = 3
So, n = 4p = (4)(3) = 12

The positive EVEN divisors of 12 are: 2, 4, 6 and 12
So, there are FOUR even divisors of n.

Answer: C

Cheers,
Brent
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Bunuel
MensaNumber
n= 4p = 2^2*p^1
so total no. of factors: (2+1)*(1+1)= 6
total no. of odd factors, since p is odd as it is a prime>2: p^1 and p^0 = 2
Total no. of even factors: 6 - 2 = 4

Now if n was n=4pq where p and q are both prime no.s greater than 2 then:
total no. of factors: (2+1)*(1+1)*(1+1)= 12
total no. of odd factors, since p is odd as it is a prime>2: (1+1)*(1+1)= 4
Total no. of even factors: 12 - 4 = 8

Hi Bunuel, could you validate my logic pls?

Yes, your approach is correct.

Bunuel in case of n=4pq
Should not there be condition that p and Q must be " distinct" prime numbers and both greater than 2.
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Walkabout
If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ?

(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight

Number of even divisors.

Since p is prime>2, p on its own is not an even divisor.

So the possibilities are 4p, 2p, 4, 2. Answer is C
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2, 4, 2p & 4p
Picking numbers drains brain energy very fast I try to avoid it as much as possible
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