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by Bill@VeritasPrep » Tue Apr 10, 2012 9:16 am
1. h(100) = 2 * 4 * 6 * 8...* 100. Since all terms are even, we can factor out a 2 from each term, leaving us with:

h(100)=2^50*(1 * 2 * 3 * 4 * 5...50). We can see that all prime numbers from 2 to 50 are already factors. Since p is h(100 + 1, p and h(100) are consecutive integers, which share no factors greater than 1. Thus, the smallest prime factor of p must be greater than 50.
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by Bill@VeritasPrep » Tue Apr 10, 2012 9:20 am
2. Starting ratio: Kaye:Alberto = 5x:3x

Change: K - 10, A + 10

Ending ratio: Kaye:Alberto = 7y:5y

We can set up individual equations for Kaye and Alberto:

Kaye: 5x - 10 = 7y

Alberto: 3x + 10 = 5y

Since the question asks about the stamps they have AFTER the change, we should solve for y. I like the combination method, so I multiplied the first equation by 3 and the second equation by -5:

15x - 30 = 21y
-15x - 50 = -25y

-80 = -4y
20=y

Kaye has 7y stamps and Alberto has 5y stamps. The difference is 2y, which equals 40.
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by Bill@VeritasPrep » Tue Apr 10, 2012 9:24 am
3. a(1) = 1, a(2) = -1, a(3)=3, a(4)=-3...

We can see that consecutive terms cancel each other out, so for the sequence to be positive, it must end with an odd term.

S1. This tells us directly that n is odd, which mets the condition we identified from the stem. Sufficient.

S2. If a(n) is positive, it must be odd. We know that any even term will be negative. Since a(n) is positive, the sum must be positive as well. Sufficient.
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by Bill@VeritasPrep » Tue Apr 10, 2012 9:28 am
4. Malik's recipe for 4 servings of a certain dish requires 3/2 cups of pasta. According to this recipe, what is the
number of cups of pasta that Malik will use the next time he prepares this dish?
To make things easier, I converted the ratio to whole numbers. Servings:Pasta = 8:3. To answer, we'll need some information about the number of servings Malik will be making.

(1) The next time he prepares this dish, Malik will make half as many servings as he did the last time he
prepared the dish.
Without knowing how many servings he made last time, this doesn't help us much. Insufficient.

(2) Malik used 6 cups of pasta the last time he prepared this dish.

We don't know how last time compares to next time. Insufficient.

When combined, we know that Malik will make half of the servings he made last time. He used 6 cups of pasta last time, so he will use 3 cups of pasta next time. Sufficient.
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by klmehta03 » Tue Apr 10, 2012 10:02 am
But for S.1 if k=-1, than a(-1)= -1; if k=-3, than a(-3)=-3; k=1, than a(1)=1 sum of all will be 0 but if we have just positive odd integers than the sum is positive. to me S.1 is insufficeint.

IMO C , OA pls?



Bill@VeritasPrep wrote:3. a(1) = 1, a(2) = -1, a(3)=3, a(4)=-3...

We can see that consecutive terms cancel each other out, so for the sequence to be positive, it must end with an odd term.

S1. This tells us directly that n is odd, which mets the condition we identified from the stem. Sufficient.

S2. If a(n) is positive, it must be odd. We know that any even term will be negative. Since a(n) is positive, the sum must be positive as well. Sufficient.
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by Bill@VeritasPrep » Tue Apr 10, 2012 6:11 pm
klmehta03 wrote:But for S.1 if k=-1, than a(-1)= -1; if k=-3, than a(-3)=-3; k=1, than a(1)=1 sum of all will be 0 but if we have just positive odd integers than the sum is positive. to me S.1 is insufficeint.

IMO C , OA pls?
Given that the sequence starts with a(1), we can't have a(-1) or other negative terms.
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by GMATGuruNY » Tue Apr 10, 2012 6:59 pm
The functions problem tests the concept of COPRIMES. I posted an explanation here:

https://www.beatthegmat.com/functions-t83704.html

I posted an alternate approach to the ratio problem here:

https://www.beatthegmat.com/number-of-st ... 02294.html
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