GMAT PREP NUMBER PROPERTY

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pkw209: Master | Next Rank: 500 Posts; Posts: 266; Joined: Mon Oct 19, 2009 9:46 pm; Thanked: 8 times; GMAT Score:690

by pkw209 » Mon Jan 11, 2010 1:44 pm

Hi,

Answer is D. Took this from Zuleron's list of 198 gmat prep questions. Thanks!

130) Positive integer k has exactly 2 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?

a. 3^2 is a factor of k
b. 7^2 is not a factor of k

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Source: Beat The GMAT — Data Sufficiency | Mon Jan 11, 2010 1:44 pm

papgust: Community Manager; Posts: 1537; Joined: Mon Aug 10, 2009 6:10 pmFollowed by: 252 members; Thanked: 653 times

by papgust » Mon Jan 11, 2010 7:41 pm

K has 2 positive prime factors - 3 and 7.
Assume that number of 3's as P and number of 7's as Q
Then, total number of factors is (P+1) * (Q+1)

To have total number of factors as 6, consider these 2 cases
(1) 2 3's and 1 7's
(2+1) * (1+1) = 6
OR
(2) 1 3's and 2 7's
(1+1) * (2+1) = 6

Let's go to each statements

A. 3^2 is a factor of K

So, there are 2 3's and 1 7's [Case (1)]. Value of k = 3*3*7 = 63
Sufficient.

B. 7^2 is not a factor of k

So, there is only 1 7's and 2 3's [Again Case (1)]. Value of k = 3*3*7 = 63
Sufficient.

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pkw209: Master | Next Rank: 500 Posts; Posts: 266; Joined: Mon Oct 19, 2009 9:46 pm; Thanked: 8 times; GMAT Score:690

by pkw209 » Wed Jan 13, 2010 11:38 am

Thanks but how did you know to consider those two cases?

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Testluv: GMAT Instructor; Posts: 1302; Joined: Mon Oct 19, 2009 2:13 pmFollowed by: 164 members; Location: Toronto; Thanked: 539 times; GMAT Score:800

by Testluv » Wed Jan 13, 2010 7:52 pm

pkw209 wrote:Hi,

Answer is D. Took this from Zuleron's list of 198 gmat prep questions. Thanks!

130) Positive integer k has exactly 2 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?

a. 3^2 is a factor of k
b. 7^2 is not a factor of k

Prime numbers are the "primary" building blocks of the composite (ie, non-prime) numbers. In other words, the composite numbers are "composed" of combinations of prime numbers.

Step 1 of the Kaplan method for data sufficiency tells us to focus on the question stem, make deductions, and determine what info is needed to answer the question:

The question stem tells us that k is composed of exactly 2 primes: 3 and 7. If k has 6 positive factors, then the factors are:

1, k, 3, 7, and 3*7

So, from the stem we can deduce that there is only one unknown positive factor, and that it is some combination of 3s or 7s.

Further, we can deduce that the missing factor has to be either 3^2 or 7^2 because if it were, say, 3^3, then 3*3*7 would also be a factor, and we would have more than six positive factors.

(1) directly tells us the unknown positive factor: 3^2 or 9.

Therefore, k = 3*3*7, and the first statement is sufficient by itself.

(2) tells us that 7^2 is not a factor of k. Therefore, the missing factor is 3^2, and the second statement is also sufficient by itself.

Both statements sufficient by themselves; choose D.

Kaplan Teacher in Toronto

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GhassanMBA: Junior | Next Rank: 30 Posts; Posts: 14; Joined: Thu Oct 08, 2009 12:34 am

by GhassanMBA » Fri Jan 15, 2010 6:51 am

Hello all,
I was trying to solve this problem and also had difficulties. After reading the proposed solutions posted here I still have some questions:

Testluv,
Why is it that if 3^3 is a factor then 3*3*7 is also a factor? A more broad question is why does the last factor HAVE to either be 3^2 or 7^2?

papgust,
I, just as pkw209 was, am also wondering why you considered just those two cases.

I hope someone will reply, it's my first time posting on this site,
thanks so much

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Testluv: GMAT Instructor; Posts: 1302; Joined: Mon Oct 19, 2009 2:13 pmFollowed by: 164 members; Location: Toronto; Thanked: 539 times; GMAT Score:800

by Testluv » Fri Jan 15, 2010 10:23 am

Why is it that if 3^3 is a factor then 3*3*7 is also a factor? A more broad question is why does the last factor HAVE to either be 3^2 or 7^2?

If there were 3 3s, we would have the following positive factors:

1, 3, 7, 3*7, 3*3, 3*3*3, 3*3*7, and k = 3*3*3*7

and that's more than six positive factors.

All positive factors of a number include all the combinations that can be made from its prime factors. As I said in my first post, the composite numbers are "composed" of combinations of prime factors.

Kaplan Teacher in Toronto

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GhassanMBA: Junior | Next Rank: 30 Posts; Posts: 14; Joined: Thu Oct 08, 2009 12:34 am

by GhassanMBA » Wed Jan 20, 2010 11:52 am

Thanks so much!

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sachindia: Master | Next Rank: 500 Posts; Posts: 258; Joined: Fri Jun 22, 2012 4:39 amFollowed by: 3 members; Location: Bengaluru, India; Thanked: 6 times; GMAT Score:640

by sachindia » Thu Nov 22, 2012 6:01 pm

I did not understand the following in testluv's post

Therefore, k = 3*3*7, and the first statement is sufficient by itself.

Why is k= only 3 multiplied by 3 multiplied by 3 multiplied by 7

why is 21 not multiplied?

Regards,
Sach

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GaneshMalkar: Senior | Next Rank: 100 Posts; Posts: 77; Joined: Tue Jul 31, 2012 6:53 amFollowed by: 1 member; Thanked: 8 times

by GaneshMalkar » Fri Nov 23, 2012 12:02 am

Let me try to put it by my way....

Will start with an example : Suppose we have number 48...when we write in standard form it is :- 2^4 * 3

To get all the factors of 48, we can apply this formula

(2^0 + 2^1 + 2^2 + 2^3 + 2^4) (3^0 + 3^1)

By Expanding we get :-

(1*1, 1*3, 2*1, and so on) all factors in ascending orders are

(1, 2, 3, 4, 6, 8, 12, 16, 24, 48)

This factors are got by multiplying each number with each other number.

If we observe the above set the number will get by multiplying first with last, second with second last(2 * 24 = 48) and this will always be in pairs...

So in our example we have

1,3,7,3*7,k

so here applying the same logic

k being the number which is 1* k

and as total 6 factors we can infer multiplying second with fifth number will give the number 63

so the missing number is 7 * 9(which is 3^2) which will fetch us the required condition

Statement 1 = 3^2 which we want sufficient

Statement 2 = not 7^2 and no other factor than 3 so it will be 3^2 which is again what we want

Answer D

Hope this help...

If you cant explain it simply you dont understand it well enough!!!
- Genius

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