GMAT PREP NUMBER PROPERTY

[pkw209]

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

[papgust]

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.

[pkw209]

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

[Testluv]

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.

[GhassanMBA]

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

[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?

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.

[GhassanMBA]

Thanks so much!

[sachindia]

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?

[GaneshMalkar]

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...