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Length of an integer

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Length of an integer

by paridhi » Fri Jul 30, 2010 8:22 am
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

OA is D
Please explain...

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by Brian@VeritasPrep » Fri Jul 30, 2010 9:51 am
Hey Paridhi,

Great question, and one that I think brings up a great strategic point. If a question asks you to maximize something (the number of factors, etc.), think strategically - here, the smaller the prime factor, the more of them we can use (if you use, say, 13, you can't multiply it very far before getting to 1,000, but if you use 2, you can repeat it several times before getting there).

Having that strategic vision going into a problem like this is helpful. We want to maximize the number of factors, which means we want to minimize the value of each factor in order to fit more. That leaves us with 2 - we can get more 2s multiplied together in a smaller overall value than anything else.

With that in mind, we just need to get a feel for how many 2s each of x and y can have before x+3y would hit 1,000. Just listing out the exponents of 2 can get us there:

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 takes us over 1,000, so this is the list of usable numbers. We want x to be as big as possible, since we're stuck multiplying y by 3 when we get to that x+3y < 1,000 calculation, so we can have x = 512. That gives a "length" of 9.

We can't fit 256*3 into the 487 we have remaining to stay under 1,000, so that means that the highest value of y we can use is 128, with a length of 7. That gives us a length of 16.

You may want to try other calculations of x and y to see if you can get something greater, but keep this in mind - we're stuck multiplying y by 3 in that x+3y calculation, so it's in our best interest to get the most out of x and to do our best with y, since we can multiply x by all 2s, but y by 2s and then one 3. That 3 is our limiter, so, again thinking strategically, we want to maximize x and then do what we can with y.
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by sanju09 » Sat Jul 31, 2010 4:10 am
Brian@VeritasPrep wrote:Hey Paridhi,

Great question, and one that I think brings up a great strategic point. If a question asks you to maximize something (the number of factors, etc.), think strategically - here, the smaller the prime factor, the more of them we can use (if you use, say, 13, you can't multiply it very far before getting to 1,000, but if you use 2, you can repeat it several times before getting there).

Having that strategic vision going into a problem like this is helpful. We want to maximize the number of factors, which means we want to minimize the value of each factor in order to fit more. That leaves us with 2 - we can get more 2s multiplied together in a smaller overall value than anything else.

With that in mind, we just need to get a feel for how many 2s each of x and y can have before x+3y would hit 1,000. Just listing out the exponents of 2 can get us there:

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 takes us over 1,000, so this is the list of usable numbers. We want x to be as big as possible, since we're stuck multiplying y by 3 when we get to that x+3y < 1,000 calculation, so we can have x = 512. That gives a "length" of 9.

We can't fit 256*3 into the 487 we have remaining to stay under 1,000, so that means that the highest value of y we can use is 128, with a length of 7. That gives us a length of 16.

You may want to try other calculations of x and y to see if you can get something greater, but keep this in mind - we're stuck multiplying y by 3 in that x+3y calculation, so it's in our best interest to get the most out of x and to do our best with y, since we can multiply x by all 2s, but y by 2s and then one 3. That 3 is our limiter, so, again thinking strategically, we want to maximize x and then do what we can with y.

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by Brian@VeritasPrep » Mon Aug 02, 2010 8:09 am
Hey, thanks, Sanju - my only regret is that vocabulary/analogies are not covered on the GMAT for you, since it sounds like you have most of them covered! I'll do my best to live up to those...
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by sanju09 » Tue Aug 03, 2010 5:59 am
Brian@VeritasPrep wrote:Hey, thanks, Sanju - my only regret is that vocabulary/analogies are not covered on the GMAT for you, since it sounds like you have most of them covered! I'll do my best to live up to those...
your most welcome Brian :)
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by showbiz » Tue Aug 03, 2010 6:11 pm
Brian@VeritasPrep wrote:Hey Paridhi,

Great question, and one that I think brings up a great strategic point. If a question asks you to maximize something (the number of factors, etc.), think strategically - here, the smaller the prime factor, the more of them we can use (if you use, say, 13, you can't multiply it very far before getting to 1,000, but if you use 2, you can repeat it several times before getting there).

Having that strategic vision going into a problem like this is helpful. We want to maximize the number of factors, which means we want to minimize the value of each factor in order to fit more. That leaves us with 2 - we can get more 2s multiplied together in a smaller overall value than anything else.

With that in mind, we just need to get a feel for how many 2s each of x and y can have before x+3y would hit 1,000. Just listing out the exponents of 2 can get us there:

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 takes us over 1,000, so this is the list of usable numbers. We want x to be as big as possible, since we're stuck multiplying y by 3 when we get to that x+3y < 1,000 calculation, so we can have x = 512. That gives a "length" of 9.

We can't fit 256*3 into the 487 we have remaining to stay under 1,000, so that means that the highest value of y we can use is 128, with a length of 7. That gives us a length of 16.

You may want to try other calculations of x and y to see if you can get something greater, but keep this in mind - we're stuck multiplying y by 3 in that x+3y calculation, so it's in our best interest to get the most out of x and to do our best with y, since we can multiply x by all 2s, but y by 2s and then one 3. That 3 is our limiter, so, again thinking strategically, we want to maximize x and then do what we can with y.
Wonderful explanation. This might be a little out of scope here, but it took me close to a minute just to understand what is going on. Any strategies to reduce the time spent reading these verbose questions?

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by Brian@VeritasPrep » Fri Aug 06, 2010 9:53 am
Hey showbiz,

Sorry - just saw this now so I apologize for not getting back for a few days!

Good question on getting started on problems like these. To me, two of the most important things to do in the first 30 seconds of reading a problem are:

1) Figure out which major concepts they're testing. The GMAT is a standardized test, so it tests the same things over and over again. Fairly immediately, I like to figure out which main themes are involved - here, they mention "prime factors", so I know that it's a factor problem, and I can gear my mind toward thinking in terms of small, prime numbers as building blocks of larger numbers. Now that I'm mentally calibrated to that, I can start to think strategically about:

2) Determine the parameters of the information, which usually means writing down any equations (or inequalities) that they give you. Essentially, you want to get information off the screen and on to your noteboard so that you can get to work on your own terms.

Here, that inequality x + 3y < 1000 is important information, so you want to have that written down and ready for use. The rest of the information sets up ground rules - because I know that we're working with prime factors, the facts that x > 1 and y > 1 I can look at as rules/definitions. They're just saying that we're only dealing with positive integers and that 1 is not a possibility (because it doesn't have any prime factors).

Learn which types of statements are "actionable" and which tend to be more "conceptual" or more rule-based. When you see statements like x > 0 or y > 1, they're not really "actionable" algebra...they're more just giving you parameters for the rest of the question so you can think of them more conceptually (x is positive, for example). the whole bit defining the length of 24, also - that's giving you a definition and example of "length", so you won't use it as algebra, but you'll want to know what "length" means.

So, in the first 30 seconds or so of reading, you should take "actionable" equations or inequalities and have them written down so that you can get to work on them; you should be aware of the rules-of-the-game and conceptual information they've given you; and you should have a direction in your mind of where the question is taking you based on the types of skills that it uses. If you train yourself to look for those things when you start reading, you should be able to get started quickly and make progress from there.

To help with that, I suggest a drill I call "Quick First Step" - take a set of 10 questions and give yourself 30 seconds on each to get started on paper, then move on to the next one. When you're done with each half-minute, go back and finish the questions. That should train you to at least get started with some action items quickly, and will also help you to determine where you tend to make mistakes in starting up a question.
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by showbiz » Fri Aug 06, 2010 11:51 am
I like that "take 10 questions" strategy. I'll do it right away and come back with a feedback. Wonderful suggestions. Thank you for your response :)

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by beatthegmatinsept » Sun Aug 29, 2010 3:51 pm
Amazing explanation @ Brian!!
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