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If \(p\) is a positive integer, is \(p\) a prime number? (1) \(p\) and \(p+1\) have the same number of factors. (2) \(p1\) is a factor of \(p\).
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16 Sep 2014, 01:45
Official Solution:If \(p\) is a positive integer, is \(p\) a prime number? (1) \(p\) and \(p+1\) have the same number of factors. Primes have 2 factors, 1 and itself, ( the reverse is also true: if a positive integer has 2 factors, then it must be a prime). So, for the answer to the question to be YES, both \(p\) and \(p+1\) must be primes. Are there consecutive primes? Yes, 2 and 3. Could we have a case when \(p\) and \(p+1\) have the same number of factors, and \(p\) is NOT a prime? Yes. For example, both 14 (not a prime) and 15 have four factors. Also, both 21 (not a prime) and 22 have four factors. Not sufficient. (2) \(p1\) is a factor of \(p\). \(p1\) and \(p\) are consecutive integers. Consecutive integers do not share any common factor but 1. Therefore, for \(p1\) to be a factor of \(p\), \(p1\) must be 1, which makes \(p\) equal to prime number 2. Sufficient. Answer: B
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Re: M3014
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30 Sep 2014, 16:26
Bunuel, can you please advise with 2 mins/question how do we quickly determine that 14 and 15 or 21 and 22 could satisfy condition A? Although I did it correctly, I took more than 4 mins to solve this question.



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05 Jun 2015, 00:06
p2bhokie wrote: Bunuel, can you please advise with 2 mins/question how do we quickly determine that 14 and 15 or 21 and 22 could satisfy condition A? Although I did it correctly, I took more than 4 mins to solve this question. yes this is a valid question..please explain sir.



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Re: M3014
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05 Jun 2015, 04:54
harshalnamdeo88 wrote: p2bhokie wrote: Bunuel, can you please advise with 2 mins/question how do we quickly determine that 14 and 15 or 21 and 22 could satisfy condition A? Although I did it correctly, I took more than 4 mins to solve this question. yes this is a valid question..please explain sir. You should spend some time and TEST values.
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Re: M3014
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10 Jan 2016, 13:12
Bunuel wrote: Official Solution:
If \(p\) is a positive integer, is \(p\) a prime number? (1) \(p\) and \(p+1\) have the same number of factors. Primes have 2 factors, 1 and itself, (the reverse is also true: if a positive integer has 2 factors, then it must be a prime). So, for the answer to the question to be YES, both \(p\) and \(p+1\) must be primes. Are there consecutive primes? Yes, 2 and 3. Could we have a case when \(p\) and \(p+1\) have the same number of factors, and \(p\) is NOT a prime? Yes. For example, both 14 (not a prime) and 15 have four factors. Also, both 21 (not a prime) and 22 have four factors. Not sufficient. (2) \(p1\) is a factor of \(p\). \(p1\) and \(p\) are consecutive integers. Consecutive integers do not share any common factor but 1. Therefore, for \(p1\) to be a factor of \(p\), \(p1\) must be 1, which makes \(p\) equal to prime number 2. Sufficient.
Answer: B What if P=1 than P1=0 won't this make statement 2 insufficient?



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Re: M3014
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10 Jan 2016, 13:14
rhio wrote: Bunuel wrote: Official Solution:
If \(p\) is a positive integer, is \(p\) a prime number? (1) \(p\) and \(p+1\) have the same number of factors. Primes have 2 factors, 1 and itself, (the reverse is also true: if a positive integer has 2 factors, then it must be a prime). So, for the answer to the question to be YES, both \(p\) and \(p+1\) must be primes. Are there consecutive primes? Yes, 2 and 3. Could we have a case when \(p\) and \(p+1\) have the same number of factors, and \(p\) is NOT a prime? Yes. For example, both 14 (not a prime) and 15 have four factors. Also, both 21 (not a prime) and 22 have four factors. Not sufficient. (2) \(p1\) is a factor of \(p\). \(p1\) and \(p\) are consecutive integers. Consecutive integers do not share any common factor but 1. Therefore, for \(p1\) to be a factor of \(p\), \(p1\) must be 1, which makes \(p\) equal to prime number 2. Sufficient.
Answer: B What if P=1 than P1=0 won't this make statement 2 insufficient? 0 is not a factor of any integer.
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Bunuel wrote: harshalnamdeo88 wrote: p2bhokie wrote: Bunuel, can you please advise with 2 mins/question how do we quickly determine that 14 and 15 or 21 and 22 could satisfy condition A? Although I did it correctly, I took more than 4 mins to solve this question. yes this is a valid question..please explain sir. You should spend some time and TEST values. You can at least think about the rule that the number of factors is the multiplication of the possibilities of the powers of its prime factors. e.g. \(14=2^1*7^1\), power possibilities are \(2^0, 2^1\) and \(7^0, 7^1\), which multiplicates in \(2*2=4 factors\) e.g. \(15=3^1*5^1\), power possibilities are \(3^0, 3^1\) and \(5^0, 5^1\), which multiplicates in \(2*2=4 factors\)



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Re: M3014
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05 Mar 2018, 03:42
Hi Bunuel,
Can you recap the rules for 0 again please?
I also got this question wrong because I thought of p=1 and p1 = 0 for statement 2.
So in effect 0 is a multiple of all numbers but not a factor of any number?
Best wishes,
Tosin



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Re: M3014
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05 Mar 2018, 04:28
ttaiwo wrote: Hi Bunuel,
Can you recap the rules for 0 again please?
I also got this question wrong because I thought of p=1 and p1 = 0 for statement 2.
So in effect 0 is a multiple of all numbers but not a factor of any number?
Best wishes,
Tosin 0 is not a factor of any integer: division by 0 is not allowed. 0 is a multiple of every integer: 0/integer = 0.
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Re: M3014
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06 Mar 2018, 05:20
If \(p\) is a positive integer, is \(p\) a prime number?
(1) \(p\) and \(p+1\) have the same number of factors.
Let p =2 & p+1 =3.............P is prime..............Answer is Yes
Let P = 21 & p+1=22..........P is Not Prime........Answer is NO
( For clarification: factors of 21: 1,3,7,21 & factors of 22: 1,2,11,22)..the each have 4 factors)
Insufficient (2) \(p1\) is a factor of \(p\).
This means that the number before p is a factor of P. This happens only in one case when P = 2.
Then P is prime = 2
Sufficient
Answer: B
Note: Two consecutive numbers have no common factor except 1.










