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.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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