GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 26 Apr 2019, 08:58

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

An even positive integer 'x' has 'y' positive integral facto

Author Message
TAGS:

Hide Tags

Senior Manager
Status: Up again.
Joined: 31 Oct 2010
Posts: 487
Concentration: Strategy, Operations
GMAT 1: 710 Q48 V40
GMAT 2: 740 Q49 V42
An even positive integer 'x' has 'y' positive integral facto  [#permalink]

Show Tags

13 Jan 2011, 03:39
3
6
00:00

Difficulty:

55% (hard)

Question Stats:

60% (02:07) correct 40% (01:37) wrong based on 286 sessions

HideShow timer Statistics

An even positive integer 'x' has 'y' positive integral factors including '1' and the number itself. How many positive integral factors does the number 4x have?

A. 4y

B. 3y

C. 16y

D. 5y

E. Cannot be determined

_________________
My GMAT debrief: http://gmatclub.com/forum/from-620-to-710-my-gmat-journey-114437.html
Intern
Joined: 30 Nov 2010
Posts: 5
WE 1: 3 years Information Technology

Show Tags

13 Jan 2011, 05:04
I tried for x=2,4 and 6. The integral factors for x and 4x did not have any specific relation.
Math Expert
Joined: 02 Sep 2009
Posts: 54696

Show Tags

13 Jan 2011, 06:15
1
1
gmatpapa wrote:
An even positive integer 'x' has 'y' positive integral factors including '1' and the number itself. How many positive integral factors does the number 4x have?

A. 4y
B. 3y
C. 16y
D. 5y
E. Cannot be determined

Probably the easiest way would be to try different numbers:
If $$x=2$$ then $$y=2$$ --> $$4x=8=2^3$$ and $$# \ of \ factors=4=2y$$;
If $$x=2^2$$ then $$y=3$$ --> $$4x=16=2^4$$ and $$# \ of \ factors=5=\frac{5y}{3}$$;
Two different answers for two values of $$x$$, hence we cannot determine the # of factors of $$4x$$.

THEORY:

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.
For more on number properties check: math-number-theory-88376.html

BACK TO THE ORIGINAL QUESTION:

Given: $$x$$ is even -->so $$x=2^p*b^q$$, where $$b$$ is some other prime factor of $$x$$ (other than 2) and $$q$$ is its power (note that x may or may not have other primes, this is just an example). The number of all factors of $$x$$ is $$y=(p+1)(q+1)$$ so:
if $$p=1$$ then $$y=2(q+1)$$;
if $$p=2$$ then $$y=3(q+1)$$;
if $$p=3$$ then $$y=4(q+1)$$;
....

Now, $$4x=2^2*x=2^{p+2}*b^q$$ and $$4x$$ will have $$(p+2+1)(q+1)=(p+3)(q+1)$$, so:
if $$p=1$$ then $$# \ of \ factors=4(q+1)=2y$$;
if $$p=2$$ then $$# \ of \ factors=5(q+1)=\frac{5y}{3}$$;
if $$p=3$$ then $$# \ of \ factors=6(q+1)=\frac{6y}{4}$$;
....

So # of factors of $$4x$$ depends on the initial power of 2 in $$x$$.

_________________
Senior Manager
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98
An even positive integer 'x' has 'y' positive integral facto  [#permalink]

Show Tags

16 Nov 2016, 03:42
1
Let’s express x as a product of its prime factors:

$$x = 2^a*3^b*5^c*…$$

Total number of its positive divisors will be

$$(a+1)*(b+1)*(c+1)*… = y$$

Now we’ll do the same for the 4x

$$4x= 2^2*2^a*3^b*5^c*… = 2^{a+2}*3^b*5^c…$$

Total number of positive divisors of 4x will be

$$(a + 3)*(b+1)*(c+1) … = y’$$

For easier calculations let’s take $$(b+1)*(c+1)*… = z$$

We have: $$(a + 3)*z = y’$$

$$(a + 1 + 2)*z = y’$$

$$(a+1)*z + 2*z = y’$$

$$y + 2*z = y’$$

So, as we can see, in order to find out the total # of positive divisors of 4x we need to know # of odd divisors of x, which is not given in the question.

Manager
Joined: 18 Jun 2017
Posts: 59
Re: An even positive integer 'x' has 'y' positive integral facto  [#permalink]

Show Tags

22 Jul 2017, 11:54
'x' has 'y' positive integral factors including '1' and the number itself.
Thus x should be a prime number.And none of the options fit in except E.
Re: An even positive integer 'x' has 'y' positive integral facto   [#permalink] 22 Jul 2017, 11:54
Display posts from previous: Sort by