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

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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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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

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Hope it helps.
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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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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?
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Re: For any integer k > 1, the term “length of an integer” refers to the n [#permalink]
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.
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Re: For any integer k > 1, the term “length of an integer” refers to the n [#permalink]
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
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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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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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Re: For any integer k > 1, the term “length of an integer” refers to the n [#permalink]
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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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Re: For any integer k > 1, the term length of an integer refers to the n [#permalink]
What is the level of this question?
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Re: For any integer k > 1, the term length of an integer refers to the n [#permalink]
jainanant909 wrote:
What is the level of this question?

You can check the difficulty level of a question, along with its stats, in the first post. For this question, the Difficulty is at a 800+ Level (the hardest you can get). The difficulty level of a question is automatically calculated based on the timer stats from the users who have attempted the question.
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Re: For any integer k > 1, the term length of an integer refers to the n [#permalink]
The term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to the integer. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3.

To find the maximum possible sum of the length of x and the length of y, given that x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, we need to find the values of x and y that maximize their lengths. One way to do this is to use the fact that the length of an integer is equal to the exponent of its prime factorization. For example, if k = 2^a * 3^b * 5^c * …, then the length of k is equal to a + b + c + …

Therefore, to maximize the length of an integer, we need to maximize its exponent. This means we need to choose the smallest possible base for each factor. Since we are dealing with positive integers, the smallest possible base is 2. So, we need to find x and y such that they are powers of 2 and satisfy the given inequality.

The largest power of 2 that is less than 1000 is 2^9 = 512. So, we can try x = 512 and see if we can find a suitable value for y. If x = 512, then x + 3y < 1000 implies that y < (1000 - 512) / 3 = 162.67. The largest power of 2 that is less than 162.67 is 2^7 = 128. So, we can try y = 128 and see if it satisfies the given inequality. If y = 128, then x + 3y = 512 + 3 * 128 = 896 < 1000. So, we have found a pair of values for x and y that satisfy the given inequality.

The length of x is equal to the exponent of its prime factorization, which is 9 in this case. The length of y is also equal to the exponent of its prime factorization, which is 7 in this case. So, the sum of the length of x and the length of y is equal to 9 + 7 = 16.

Therefore, the maximum possible sum of the length of x and the length of y is 16, when x = 512 and y = 128.
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