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
Positive integer k
-
- Senior | Next Rank: 100 Posts
- Posts: 87
- Joined: Sun Apr 19, 2009 11:07 pm
-
- Master | Next Rank: 500 Posts
- Posts: 418
- Joined: Wed Jun 11, 2008 5:29 am
- Thanked: 65 times
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.
-BM-
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.
-BM-
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.
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.
- cubicle_bound_misfit
- Master | Next Rank: 500 Posts
- Posts: 246
- Joined: Mon May 19, 2008 7:34 am
- Location: Texaco Gas Station
- Thanked: 7 times
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.
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.
Cubicle Bound Misfit
-
- Senior | Next Rank: 100 Posts
- Posts: 87
- Joined: Sun Apr 19, 2009 11:07 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.
Trust but verify.
- dmateer25
- Community Manager
- Posts: 1049
- Joined: Sun Apr 06, 2008 5:15 pm
- Location: Pittsburgh, PA
- Thanked: 113 times
- Followed by:27 members
- GMAT Score:710
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
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
-
- Master | Next Rank: 500 Posts
- Posts: 148
- Joined: Wed Dec 10, 2008 5:13 pm
- Location: SF, CA
- Thanked: 12 times
Good explanation by DaveGill (MGMAT forum)
https://www.manhattangmat.net.in/forums/post11734.html
https://www.manhattangmat.net.in/forums/post11734.html
The more you look, the more you see.
-
- Master | Next Rank: 500 Posts
- Posts: 418
- Joined: Wed Jun 11, 2008 5:29 am
- Thanked: 65 times
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.cubicle_bound_misfit wrote:
but if 3^2 is a factor 3
and for this case
x =1 y=3
In any case, dmateer25 has explained the theory behind this problem. Post back if something still bothers you.
-BM-
-
- Senior | Next Rank: 100 Posts
- Posts: 87
- Joined: Sun Apr 19, 2009 11:07 pm
-
- Senior | Next Rank: 100 Posts
- Posts: 57
- Joined: Sun Jul 07, 2013 5:35 am
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
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
-
- Senior | Next Rank: 100 Posts
- Posts: 57
- Joined: Sun Jul 07, 2013 5:35 am
Can someone help me with this doubt posted by me in previous post? Experts please help
Thanks,
Pooja
Thanks,
Pooja
GMAT/MBA Expert
- [email protected]
- Elite Legendary Member
- Posts: 10392
- Joined: Sun Jun 23, 2013 6:38 pm
- Location: Palo Alto, CA
- Thanked: 2867 times
- Followed by:511 members
- GMAT Score:800
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.
GMAT assassins aren't born, they're made,
Rich
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.
GMAT assassins aren't born, they're made,
Rich
-
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
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.
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
-
- Senior | Next Rank: 100 Posts
- Posts: 57
- Joined: Sun Jul 07, 2013 5:35 am
- sanju09
- GMAT Instructor
- Posts: 3650
- Joined: Wed Jan 21, 2009 4:27 am
- Location: India
- Thanked: 267 times
- Followed by:80 members
- GMAT Score:760
A Rule with Example: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
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]
The mind is everything. What you think you become. -Lord Buddha
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com