GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 12 Jul 2020, 11:46 ### 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

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # For any integer k > 1, the term “length of an integer” refers to the n

Author Message
TAGS:

### Hide Tags

Manager  Joined: 28 Aug 2010
Posts: 139
For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

13
132 00:00

Difficulty:   95% (hard)

Question Stats: 45% (02:34) correct 55% (02:51) wrong based on 1037 sessions

### HideShow timer Statistics

For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

A. 5
B. 6
C. 15
D. 16
E. 18
Math Expert V
Joined: 02 Sep 2009
Posts: 65194
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

57
54
ajit257 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

a. 5
b. 6
c. 15
d. 16
e. 18

Can some explain an elegant way of doing such a problem which would take less time.

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

_________________
Manager  Joined: 01 Nov 2010
Posts: 197
Location: India
Concentration: Technology, Marketing
GMAT Date: 08-27-2012
GPA: 3.8
WE: Marketing (Manufacturing)
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

11
1
here is my approach :
we know that : x > 1, y > 1, and x + 3y < 1000,
and it is given that length means no of factors.
for any value of x and y, the max no of factors can be obtained only if factor is smallest no & all factors are equal.
2^1 =2
2^2 =4
2^3=8
2^4=16
2^5=32
2^6=64
2^7=128
2^8=256
2^9=512
2^10 =1024 (opps//it exceeds 1000, so, x can't be 2^10)
so, max value that X can take is 2^9 , for which has "length of integer" is 9.
now, since x =512 , & x+3y<1000
so, 3y<488
==> y<162
so, y can take any value which is less than 162. and to get the maximum no of factors of smallest integer, we can say y=2^7
for 2^7 has length of integer is 7.

SO, combined together : 9+7 = 16.
And is D.

Hope it will help.
##### General Discussion
Manager  Joined: 28 Aug 2010
Posts: 139
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

ah ! . awesome..thanks Bunuel !
Math Expert V
Joined: 02 Sep 2009
Posts: 65194
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

3
3
Manager  Joined: 07 Sep 2010
Posts: 245
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

1
Bunuel wrote:

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Hi Bunuel,
I tried solving this question. However, I thought x and y to be different. That's why I put x= 2 and y = 3, in order to minimize the prime bases and thus maximize the powers of the primes.
Isnt the question implying x and y to be different. Otherwise the given equation x+3y<1000 is as good as x+3x<1000

..getting what I am trying to put across?
Math Expert V
Joined: 02 Sep 2009
Posts: 65194
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

imhimanshu wrote:
Bunuel wrote:

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Hi Bunuel,
I tried solving this question. However, I thought x and y to be different. That's why I put x= 2 and y = 3, in order to minimize the prime bases and thus maximize the powers of the primes.
Isnt the question implying x and y to be different. Otherwise the given equation x+3y<1000 is as good as x+3x<1000

..getting what I am trying to put across?

We are not told that x and y are distinct, so we cannot assume this.
_________________
Manager  Joined: 17 Oct 2010
Posts: 69
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

Bunuel wrote:
imhimanshu wrote:
Bunuel wrote:

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Hi Bunuel,
I tried solving this question. However, I thought x and y to be different. That's why I put x= 2 and y = 3, in order to minimize the prime bases and thus maximize the powers of the primes.
Isnt the question implying x and y to be different. Otherwise the given equation x+3y<1000 is as good as x+3x<1000

..getting what I am trying to put across?

We are not told that x and y are distinct, so we can not assume this. Next, even if we were told that they are distinct the answer still would be D: 2^8*3=768<100 also has the length of 8+1=9.

Hi Bunuel

x+3y<1000 and if x and y are distinct then shouldn't this be
2^9+3^5<1000 for a length of 14 which is not among the options, so I guess the question didn't mean that they are distinct. Please can you explain once more if they are distinct how can the answer still be 16 (D)

Thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 65194
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

Joy111 wrote:
Hi Bunuel

x+3y<1000 and if x and y are distinct then shouldn't this be
2^9+3^5<1000 for a length of 14 which is not among the options, so I guess the question didn't mean that they are distinct. Please can you explain once more if they are distinct how can the answer still be 16 (D)

Thanks

We are not told that x and y are distinct. But if we were told that, the answer would be 13 not 14: x+3y=2^9+3*3^4=755<1,000 --> 9+4=13.
_________________
Manager  Joined: 17 Oct 2010
Posts: 69
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

Bunuel wrote:
Joy111 wrote:
Hi Bunuel

x+3y<1000 and if x and y are distinct then shouldn't this be
2^9+3^5<1000 for a length of 14 which is not among the options, so I guess the question didn't mean that they are distinct. Please can you explain once more if they are distinct how can the answer still be 16 (D)

Thanks

We are not told that x and y are distinct. But if we were told that, the answer would be 13 not 14: x+3y=2^9+3*3^4=755<1,000 --> 9+4=13.

Hi Bunuel

shouldn't we take 3*3^4= 3^5 ( adding the exponents with same base) and hence 9+5= 14 ?
Math Expert V
Joined: 02 Sep 2009
Posts: 65194
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

1
Joy111 wrote:
Bunuel wrote:
Joy111 wrote:
Hi Bunuel

x+3y<1000 and if x and y are distinct then shouldn't this be
2^9+3^5<1000 for a length of 14 which is not among the options, so I guess the question didn't mean that they are distinct. Please can you explain once more if they are distinct how can the answer still be 16 (D)

Thanks

We are not told that x and y are distinct. But if we were told that, the answer would be 13 not 14: x+3y=2^9+3*3^4=755<1,000 --> 9+4=13.

Hi Bunuel

shouldn't we take 3*3^4= 3^5 ( adding the exponents with same base) and hence 9+5= 14 ?

No. Please read the question carefully "what is the maximum possible sum of the length of x and the length of y". The length of x is 9 and the length of y is 4, so the sum is 9+4=13.
_________________
Manager  Joined: 17 Oct 2010
Posts: 69
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

1
We are not told that x and y are distinct. But if we were told that, the answer would be 13 not 14: x+3y=2^9+3*3^4=755<1,000 --> 9+4=13.[/quote]

Hi Bunuel

shouldn't we take 3*3^4= 3^5 ( adding the exponents with same base) and hence 9+5= 14 ?[/quote]

No. Please read the question carefully "what is the maximum possible sum of the length of x and the length of y". The length of x is 9 and the length of y is 4, so the sum is 9+4=13.[/quote]

Awesome , thanks ,really fell in the trap for that one !! + 1
Senior Manager  Joined: 13 Jan 2012
Posts: 272
Weight: 170lbs
GMAT 1: 740 Q48 V42
GMAT 2: 760 Q50 V42
WE: Analyst (Other)
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

2
What's the source of this problem? Basically, you know that in order to get the longest "length", we'll want all 2's in the prime factorization. So how can we satisfy x + 3y < 1000 where x and y are both 2 to the nth power.

Let's start here:
2^8 + 3(2^8) = ?
256 + 768 = 1024

TOO HIGH, but that's interesting. It definitely looks like we can play around with this somehow to reach it. We know that it won't make sense to increase to 2^9 for y's value because 2^9 is 512 and 512*3 > 1000. How about the other way around?

2^9 + 3(2^7) = ?
512 + 3(128) = 896

9 + 7 = 16.

We know we won't be able to get much higher than that because 2^10 as x is the only other move we could try and that is > 1000 by itself. (It's 1024 which you should have memorized!) So answer is D = 16.

Hope that helps.
Math Expert V
Joined: 02 Sep 2009
Posts: 65194
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Number Properties: math-number-theory-88376.html

All DS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38
All PS Number Properties Problems to practice: search.php?search_id=tag&tag_id=59

_________________
Intern  Joined: 04 Nov 2012
Posts: 47
Schools: NTU '16 (A)
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

1
Bunuel wrote:
ajit257 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

a. 5
b. 6
c. 15
d. 16
e. 18

Can some explain an elegant way of doing such a problem which would take less time.

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Hi Bunuel, Don't we need to check for the other case. i.e when try to maximise the length of Y rather than that of X??
Manager  Joined: 26 Jul 2012
Posts: 57
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

1
12bhang wrote:
Bunuel wrote:
ajit257 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

a. 5
b. 6
c. 15
d. 16
e. 18

Can some explain an elegant way of doing such a problem which would take less time.

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Hi Bunuel, Don't we need to check for the other case. i.e when try to maximise the length of Y rather than that of X??

I think we can try both cases to see which gives you the maximum Length.

1) maximize Y

x + 3 y < 1000
2^9 = 512
2 ^8 = 256
so y = 2^ 8
and now x is 999 - 3(256) = 231
x can be 2^7 = 128
So total length = 8 + 7 = 15

2) maximize x

x + 3 y < 1000
2^9 = 512
2 ^ 10 = 1024
length of x = 9
now, y = (999 - 512 ) / 3 = 162.x
or y is 2^7

Total length is 9+7 = 16.

Now, you might want to NOT do the maximization of y because you know that most 2s will be in x and NOT y. For example, we have to first realize that we want the number with the MOST 2s in it. x can be that number as illustrated by this example: If y = 512 -> 512 * 3 > 1000 and if x = 512 , well x can be 512. So, we would get a length of 9 out of it.

If we were doing, x + y < 1000 then the lengths can be inter-changed; however, because of the 3 next to y, we know that the length of y will have to be LESS than the length of x.

Senior Manager  Joined: 07 Apr 2012
Posts: 311
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

Bunuel wrote:
ajit257 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

a. 5
b. 6
c. 15
d. 16
e. 18

Can some explain an elegant way of doing such a problem which would take less time.

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Hi Bunuel,

What is the logic?
Intern  B
Joined: 27 Apr 2015
Posts: 32
Location: India
GMAT 1: 730 Q50 V40
WE: Operations (Telecommunications)
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

Bunuel wrote:
ajit257 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

a. 5
b. 6
c. 15
d. 16
e. 18

Can some explain an elegant way of doing such a problem which would take less time.

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Bunuel, i must say, u rock man!!!
Manager  Joined: 22 Feb 2016
Posts: 80
Location: India
Concentration: Economics, Healthcare
GMAT 1: 690 Q42 V47
GMAT 2: 710 Q47 V39
GPA: 3.57
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

definitely an extremely tough question.

Now the question basically boils down to the given fundamental fact that length of an interger is dependent on the power of the prime factor

So if we want maximum length we need minimum pf base and if need minimum length we need maximum pf base. (if the question was asked other way round)

hence we need to understand the maximization and minimization techniques well.

x+3y=999 (since we need the maximum possible value in this case)

let us assume the base PF is 2 (the lowest possible PF) 2^9=512
3y=999-512

y= 162

again assuming the base value of y as 2. Run the drill 2*2*2*2...2^7=128 closest to our answer (remember 999 is not the actually value we have assumed it as the best possible max value)

so we have x=2^9
y=2^7

Lx+ly= 9+7=16
YAY :D
Current Student B
Joined: 09 Sep 2014
Posts: 21
Location: Georgia
GMAT 1: 610 Q47 V27
GPA: 3.24
Re: For any integer k > 1, the term “length of an integer” refers to the n  [#permalink]

### Show Tags

ajit257 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

A. 5
B. 6
C. 15
D. 16
E. 18

Hi everyone, I have read all comments below the problem but still cannot understand why the max. possible sum is 16.
We all agree that x=512=2^9 so length of x is 9, great!
Now, we are left with 3*y that could be 3*2^7=3*128=384. So, still 512+384<1000, right?. BUT length of y is 8, am i wrong? 384=2x2x2x2x2x2x2x3 8 primes!
so max. possible sum appears to be 9+8=17 which is not in the list of answer choices, so I guess I would go whatever the other choice I have but why everyone is posting that length of y is 7?
Maybe I miss something after sleepless night, doing math test whole night, I don't know Thanks Re: For any integer k > 1, the term “length of an integer” refers to the n   [#permalink] 07 Dec 2016, 19:09

Go to page    1   2    Next  [ 22 posts ]

# For any integer k > 1, the term “length of an integer” refers to the n   