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30 Dec 2019, 07:10
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
34% (02:05) correct
66%
(01:56)
wrong
based on 96
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Q. p is a natural number, is \((8p+1)\) prime?
Statement 1: p is prime
Statement 2: \((8p-1)\) is prime
Statement 1: p is prime
Statement 2: \((8p-1)\) is prime
30 Dec 2019, 14:42
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uchihaitachi
First off, let's look at the given info. p is a natural number, which means p is one of the "counting numbers": 1, 2, 3, 4, etc. (By the way, on the GMAT I think you'd be more likely to see "positive integer" than "natural number," for the sake of clarity.)
Is 8p + 1 prime? I'll have to test cases for p, then check whether 8p + 1 is prime or not.
Statement 1: p is prime, so we can only test prime numbers. Let's just start with the smallest prime (p = 2) and see what happens.
p = 2 -> 8p + 1 = 8(2) + 1 = 17, which is prime. YES
p = 3 -> 8p + 1 = 8(3) + 1 = 25, which is NOT prime. NO
We have a yes and a no, so the statement is insufficient.
Statement 2: 8p - 1 is prime. Interesting. I don't really want to guess and test a ton of values of p just to figure out whether they fit the statement in the first place. Instead, I'm going to start with the statement, instead of trying to guess the right value of p.
If 8p - 1 is prime, then suppose that 8p - 1 = 2. Then, 8p = 3. However, then p isn't a natural number. Never mind! Try a bigger value.
Suppose that 8p - 1 = 23. Then 8p = 24, and p = 3. So that's a good case! In this case, 8p + 1 = 25, which is NOT prime. NO
Now, suppose that 8p - 1 = 31. Then 8p = 32, and p = 4. So that's also a good case! (Remember, we don't know that p is prime in this statement.) In this case, 8p + 1 = 8(4) + 1 = 33, which is NOT prime. NO
Hm. Interesting. Is there a case where 8p + 1 IS prime? Try one more:
8p - 1 = 47. So, 8p = 48, and p = 6. That's a good case. In this case, 8p + 1 = 8(6) + 1 = 49, which is NOT prime. NO
At this point on test day, I'd stop and assume that the statement was sufficient, since it seems to always give me non-primes.
But, because we're studying and not taking a real test, let's actually try to prove for sure whether it's sufficient or insufficient. To generate the cases above, I was thinking of primes that were 1 less than a multiple of 8. That's because p has to be a natural number, and those are the only primes values for 8p - 1 that will actually cause p to come out as a round number.
So what we're really saying is, if a number 1 less than a multiple of 8 is prime, then is the number 1 greater than that multiple of 8 also prime? To get a handle on it, I'm going to start jotting down multiples of 8. I'm using a P for primes, and an X for non-primes.
8(1) = 8 | 8-1 = 7 P | 8 + 1 = 9 x
8(2) = 16 | 16-1 = 15 x | don't have to keep testing, since 8p-1 isn't prime.
8(3) = 24 | 24 - 1 = 23 P | 24 + 1 = 25 x
8(4) = 32 | 32 - 1 = 31 P | 32 + 1 = 33 x
8(5) = 40 | 40 - 1 = 39 x | don't have to keep testing, since 8p-1 isn't prime.
8(6) = 48 | 48 - 1 = 47 P | 48 + 1 = 49 x
8(7) = 56 | 56 - 1 = 55 x | don't have to keep testing, since 8p-1 isn't prime.
8(8) = 64 | 64 - 1 = 63 x | don't have to keep testing, since 8p-1 isn't prime.
8(9) = 72 | 72 - 1 = 71 P | 72 + 1 = 73 P
Phew! I found a pair that works with the statement, and finally gives an answer of 'yes' to the question.
If p = 9, 8p - 1 = 71, and 8p + 1 = 73, then 8p - 1 is prime and the answer to the question is 'yes'. Other values, like p = 3, 4, or 6, give us answers of 'no'.
This statement is insufficient. If anybody knows a faster way to prove that, I'm very interested in hearing it!
Statements 1 and 2 together
Yikes. Unfortunately, we can't take a shortcut here, because when we found a 'yes' case for statement 2, we used a value of p that wasn't prime. (If we'd used one that was prime, we could be done right now, since we'd have yes and no cases that fit both statements.)
Is there a 'yes' case for statement 2 that also fits statement 1?
On test day, I definitely wouldn't bother trying to figure that out... seems very time consuming. In fact, I'm starting to suspect that this problem would never appear on the GMAT! I cheated and searched for it online, and found this interesting proof: https://math.stackexchange.com/question ... -composite
It turns out that the right answer is C, but it's right for rather complex number-theoretic reasons that wouldn't come into play on the GMAT. Read it if you're curious...but if you're struggling with this problem and worried about it on the GMAT, don't worry, you won't see anything that requires this much case testing or this much number theory.
30 Dec 2019, 23:50
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Statement 1-
p=2; 8p+1=17
p=3; 8p+1= 25
Statement 2-
8p-1= 23; 8p+1=25
8p-1=71; 8p+1=73
Combining both statements
If both p and 8p-1 are prime, both leave 1 remainder when divided by 6
If a number is prime, it must leave either 1 or -1 as remainder when divided by 6
Case1.
p=1 mod 6; 8p-1=(7-6)mod 6=1 mod 6 {both can be prime}
Case 2
p=-1 mod 6; 8p-1=3 mod 6 (can never be prime)
p=1 mod 6; 8p-1=1 mod 6; 8p+1=3 mod 6 (can't be a prime)
Sufficient
p=2; 8p+1=17
p=3; 8p+1= 25
Statement 2-
8p-1= 23; 8p+1=25
8p-1=71; 8p+1=73
Combining both statements
If both p and 8p-1 are prime, both leave 1 remainder when divided by 6
If a number is prime, it must leave either 1 or -1 as remainder when divided by 6
Case1.
p=1 mod 6; 8p-1=(7-6)mod 6=1 mod 6 {both can be prime}
Case 2
p=-1 mod 6; 8p-1=3 mod 6 (can never be prime)
p=1 mod 6; 8p-1=1 mod 6; 8p+1=3 mod 6 (can't be a prime)
Sufficient
uchihaitachi
