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

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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
[Reveal] Spoiler: OA

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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.

Answer: D.
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ah ! . awesome..thanks Bunuel !
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New post 15 Mar 2012, 05:15
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.

Answer: D.


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?
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New post 15 Mar 2012, 06:33
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.

Answer: D.


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.
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New post 15 May 2012, 09:27
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.

Answer: D.


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)

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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.
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New post 16 May 2012, 01:19
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 ?

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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.
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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

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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.
hence, lets start with smallest no 2.
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.
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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.

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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.

Answer: D.



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??

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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.

Answer: D.



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.

Answer D.

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Re: For any integer k > 1, the term “length of an integer” refers to the n [#permalink]

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New post 11 Nov 2013, 09:17
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.

Answer: D.

Hi Bunuel,

How did you know to start with "x" and not "y"?
What is the logic?
When I start with "y" I get a different result....

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Re: For any integer k > 1, the term “length of an integer” refers to the n [#permalink]

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New post 04 Sep 2015, 06:35
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.

Answer: D.



Bunuel, i must say, u rock man!!!

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Re: For any integer k > 1, the term “length of an integer” refers to the n [#permalink]

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New post 03 Dec 2016, 21:50
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

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Re: For any integer k > 1, the term “length of an integer” refers to the n [#permalink]

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New post 07 Dec 2016, 19:09
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 :roll:
Thanks

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Re: For any integer k > 1, the term “length of an integer” refers to the n   [#permalink] 07 Dec 2016, 19:09

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