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Positive integer k

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Positive integer k

by success1111 » Mon Apr 27, 2009 11:53 pm
The positive integer K has exactly two positive prime factors 3 and 7.If K has a total of 6 positive factors including 1 and K,WHAT IS THE VALUE OF k?

1)3^2 is a factor of k

2) 7^2 is not a factor of k
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by bluementor » Tue Apr 28, 2009 12:48 am
Since we know k has exactly 7 and 3 as prime factors, we can summarize as follows:

k = (3^x)(7^y) , where x and y are positive integers.

we also know that k has exactly 6 different positive factors. so,

6 = (x+1)(y+1)

we need to know the values of x and y to evaluate k.

Statement 1: 3^2 is a factor of k

If 3^2 = 3^x, then x=2. In this case, y must equal to 1.

If x = 3 (or 4), then y will not be an integer.
If x = 5, then y will be 0. This case is also not possible since we know y must be at least 1 (because 7 is a known factor of k).

Therefore, the only possible value for x is 2. And from this we can conclude that y can only be 1. Sufficient.

Statement 2: 7^2 is not a factor of k

Therefore, y cannot be larger than 1. And since y must at least be 1 (because 7 is a known factor of k), we can conclude that y = 1. And from this we can determine x = 2. Sufficient.

Choose D.

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by gdsurdy » Tue Apr 28, 2009 11:07 am
I am not sure if I understood the question correctly. It says K has exactly two +ve prime factors 3 and 7. It has 6 factors in all including 1 and K.

Doesn't this interpret to the 6 factors being 1, 3, 7, _ , _ and K.

If we look at the question this way the answer would be E.
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by cubicle_bound_misfit » Tue Apr 28, 2009 11:59 am
Hi Blue Mnetor,

If 3^2 = 3^x, then x=2. In this case, y must equal to 1.

but if 3^2 is a factor 3
and for this case

x =1 y=3

so we get two different value of x, then HOW COME 1 IS SUFF?
PLEASE HELP.
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by success1111 » Tue Apr 28, 2009 4:48 pm
gdsurdy wrote:I am not sure if I understood the question correctly. It says K has exactly two +ve prime factors 3 and 7. It has 6 factors in all including 1 and K.

Doesn't this interpret to the 6 factors being 1, 3, 7, _ , _ and K.

If we look at the question this way the answer would be E.

FYI,this a GMATPREP question.It is a clue for you to know much more about number properties because they come in different disguise.

BM,thanks a lot for your input but i am still having trouble on how your arrive at your answer.

Cubic,thanks a well.

Experts, i need help on this question.
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by dmateer25 » Tue Apr 28, 2009 5:01 pm
Let me take a stab at this one.

The number of factors a number has is based on the prime factorization of a number.

Let’s say we have a number x.
The prime factorization of x= p^2 * p^2. The number of factors of x would be (3)(3)=9.

The rule is if x= p^n * q^m and p and q are prime, then it has (n+1)(m+1) factors.

So looking at this question we know that k has two prime factors 3 and 7. Also, we know that k has a total of 6 factors.

K can be 3^2 * 7^1 (remember we need to have 6 total factors and (2+1)(1+1)=6).

K can also be 3^1 * 7^2.


So let’s look at the statements.

1)3^2 is a factor of k
Well this is sufficient because as I stated above we only have 2 possibilities and only one has 3^2 as a factor. So K = 3^2 *7^1.

Suff

2) 7^2 is not a factor of k

This is also sufficient because we know 7^2 isn’t a factor, so 3^2 * 7^1 must equal k.

Suff

Choose D
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by lilu » Tue Apr 28, 2009 9:01 pm
Good explanation by DaveGill (MGMAT forum)
https://www.manhattangmat.net.in/forums/post11734.html
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by bluementor » Wed Apr 29, 2009 12:15 am
cubicle_bound_misfit wrote:
but if 3^2 is a factor 3
and for this case

x =1 y=3
CBM, I'm not sure I understand how you arrived at this. 3^2 cannot be a factor of 3. So this case is not valid.

In any case, dmateer25 has explained the theory behind this problem. Post back if something still bothers you.

-BM-
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by success1111 » Wed Apr 29, 2009 8:02 pm
Yeah. OA is D.
Thanks everyone for your contribution.
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by Winner2013 » Sat Feb 01, 2014 12:09 pm
I have a doubt about this question and need some help. I guess this doubt is really stupid but still I would appreciate if someone can help.

The question clearly says that the number k has exactly 2 positive prime factors - 3 and 7. now for 63- after factorization we get

63= 3*3*7

so we have 3 prime factors here right? - 3,3,7. Then how does the question say only 2 prime factors and the answer comes out to be 63?

am i interpreting the question in a wrong way? if a prime factor(3) is repeated as in case of 63, what do we say about how many prime factors does the number have?

please help.

thanks,
Pooja
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by Winner2013 » Mon Feb 03, 2014 12:04 pm
Can someone help me with this doubt posted by me in previous post? Experts please help

Thanks,
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by [email protected] » Mon Feb 03, 2014 3:04 pm
Hi Pooja,

When a question discusses the number of "prime factors" in an integer, then duplicates don't count.

For example, then number 21 and 63 have the same prime factors: 3 and 7. While 63 has "another 3", factoring (and prime factoring) questions don't count the duplicates.

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by Matt@VeritasPrep » Mon Feb 03, 2014 4:29 pm
Good question! By "prime factors", this question means "unique prime factors". This implication is VERY common on math tests and should probably be assumed unless there's a real reason to suspect otherwise.

Another detail here: if a number has exactly six factors, it MUST have either 5 identical prime factors (and no others), or 2 of prime factor p and 1 of prime factor q, where p is not equal to q. So from that deduction alone you would be able to catch the (admittedly imprecise) implication mentioned above.
Winner2013 wrote:I have a doubt about this question and need some help. I guess this doubt is really stupid but still I would appreciate if someone can help.

The question clearly says that the number k has exactly 2 positive prime factors - 3 and 7. now for 63- after factorization we get

63= 3*3*7

so we have 3 prime factors here right? - 3,3,7. Then how does the question say only 2 prime factors and the answer comes out to be 63?

am i interpreting the question in a wrong way? if a prime factor(3) is repeated as in case of 63, what do we say about how many prime factors does the number have?

please help.

thanks,
Pooja
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by Winner2013 » Thu Feb 06, 2014 1:05 pm
Thank you Matt and Rich.

Your explanations were really helpful.

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by sanju09 » Fri Feb 07, 2014 1:21 am
success1111 wrote:The positive integer K has exactly two positive prime factors 3 and 7.If K has a total of 6 positive factors including 1 and K,WHAT IS THE VALUE OF k?

1)3^2 is a factor of k

2) 7^2 is not a factor of k
A Rule with Example:

If K = 3^a.7^b, a, b ≠ 0, then K would have a total of (a + 1) (b + 1) number of factors, including 1 and K. So, if K has a total of 6 positive factors including 1 and K, then the possibilities are that either K = 3^1.7^2 or K = 3^2.7^1. So this is what we have to look to while testing the statements.

(1) If 3^2 is a factor of K, then K must be equal to 3^2.7^1 only. Sufficient

(2) If 7^2 is not a factor of K, then it must be that 7^1 is a factor of K, hence K must be equal to 3^2.7^1 only. [spoiler]Sufficient

Pick D[/spoiler]
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