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What is the value of the positive integer a?

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What is the value of the positive integer a?  [#permalink]

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New post 29 Jan 2015, 08:28
1
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A
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C
D
E

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Question Stats:

43% (01:58) correct 57% (02:21) wrong based on 131 sessions

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Re: What is the value of the positive integer a?  [#permalink]

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New post Updated on: 31 Jan 2015, 12:25
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Hi shriramvelamuri,

Yes, there is a pattern involving prime factorization that can help speed you up on this question. While it's a relatively rare pattern, the GMAT might slip in into your Test in 1 question (when you're scoring at a sufficiently high level).

For a positive integer to have EXACTLY 6 factors, the prime-factorization of that number will yield 2 DIFFERENT primes, one of which shows up TWICE.

For example:
12 = (2)(2)(3) = 2 different primes, one of which shows up twice.

Factors of 12: 1,12, 2,6, 3,4

We can use this pattern to quickly find the various integers that fit this pattern (and 'leapfrog' all of the integers that don't). Since the two Facts "hint" at the fact that we're going to be dealing with 2-digit numbers, I'm going to limit my work to those.....

Start with the prime that shows up twice:
(2^2)(3) = 12
(2^2)(5) = 20
x7 = 28
x11 = 44
x13 = 52
x17 = 68
Etc.

(3^3)(2) = 18
(3^3)(5) = 45
x7 = 63
x11 = 99

(5^2)(2) = 50
(5^2)(3) = 75

(7^2)(2) = 98

With this much smaller list of numbers, you should be able to pinpoint the possibilities that "fit" the Facts.

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Originally posted by EMPOWERgmatRichC on 29 Jan 2015, 13:28.
Last edited by EMPOWERgmatRichC on 31 Jan 2015, 12:25, edited 1 time in total.
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Re: What is the value of the positive integer a?  [#permalink]

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New post 29 Jan 2015, 11:22
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ans E..
1) it can give us various values insufficient
2) it can again give us more than 1 value for eg a=44(44 and 45 have 6 factors) and a=63(63 and 64 have 6 factors)...
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Re: What is the value of the positive integer a?  [#permalink]

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New post 29 Jan 2015, 12:45
HI Chetan,

Is there a logical short cut or a logical way for solving this question.

chetan2u wrote:
ans E..
1) it can give us various values insufficient
2) it can again give us more than 1 value for eg a=44(44 and 45 have 6 factors) and a=63(63 and 64 have 6 factors)...

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Re: What is the value of the positive integer a?  [#permalink]

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New post 29 Jan 2015, 12:52
Hi chetan2u,

You should double-check your work: 64 does NOT have 6 factors (it has 7).

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Re: What is the value of the positive integer a?  [#permalink]

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New post 29 Jan 2015, 13:03
Hi Empower,

Is there a shortcut to solve this question. Going after every number will waste our time. please educate me.


quote="EMPOWERgmatRichC"]Hi chetan2u,

You should double-check your work: 64 does NOT have 6 factors (it has 7).

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Rich[/quote]
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Re: What is the value of the positive integer a?  [#permalink]

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New post 29 Jan 2015, 18:24
EMPOWERgmatRichC wrote:
Hi chetan2u,

You should double-check your work: 64 does NOT have 6 factors (it has 7).

GMAT assassins aren't born, they're made,
Rich


hi thanks buddy...
I should have been careful as 64 is a perfect square so cannot have even number of factors...
+1 for you
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Re: What is the value of the positive integer a?  [#permalink]

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New post 29 Jan 2015, 21:57
1
Here you go:

a is a positive integer (given)

St1: a and (a+1) each have exactly 6 factors.

No limit defined on the value of a as a can have any value between 1 and infinity.

Not sufficient

St2: a < 76

a can have any value between 1 and 76
Not sufficient

Combining :

a and (a+1) each have exactly 6 factors and a < 76.

key here is : a and (a+1) are co-prime integer.

Consider p, q, r, and s be 4 different prime numbers.

for number of factors to be 6

a = p^2 * q
(a+1) = r^2 * s

as we have to find the value less than 76... substituting values will not take much of a time....

the only value comes up is

a = 44 = 2^2 * 11
(a+1) = 45 = 3^2 * 5

Hence C is sufficient.

Note: We can also use the following property :
GCD of two co-prime integers = 1
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Re: What is the value of the positive integer a?  [#permalink]

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New post 31 Jan 2015, 05:14
the answer is E

statement 1 - not sufficient - multiple answers

statement 2 - not sufficient - 2 answers 44 and 75
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Re: What is the value of the positive integer a?  [#permalink]

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New post 02 Feb 2015, 03:44
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Bunuel wrote:
What is the value of the positive integer a?

(1) a and (a+1) each have exactly 6 factors.

(2) a < 76

Kudos for a correct solution.


VERITAS PREP OFFICIAL SOLUTION:

If you look first at statement 2, you can see that it's pretty useless, as there are 76 positive integers less than 77 so statement 2 is obviously insufficient. So as you start to look at statement 1, you should begin thinking in that range of statement 2 so that you're efficient in your work.

Knowing Data Sufficiency, you may want to try the number at the exact end of the range in statement 2 - they don't pick those inequality ranges by accident! If a is 75 and a + 1 is 76, then a has six factors (1 and 75; 3 and 25; and 5 and 15), and 76, a + 1, has six factors (1 and 76, 2 and 28, 4 and 19).

In seeing those factors and thinking systematically, you might recognize that each of those factorizations includes a square (75 has 25; 76 has 4) and start looking for similar numbers further down the scale. One more such pairing is 44 and 45: 44 has a square in its factorization (4) and so does 45 (9). And each does have six factors (44: 1 and 44; 2 and 22; 4 and 11. 45: 1 and 45; 3 and 15; 5 and 9). So a could be either 44 or 75, and therefore the answer is E.
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Re: What is the value of the positive integer a?  [#permalink]

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Re: What is the value of the positive integer a?   [#permalink] 27 Mar 2019, 17:41
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