If p is a positive integer, is p a prime number?
(1) p and p+1 have the same number of factors.
(2) p−1 is a factor of p.
Answer : B
If p is a positive integer, is p a prime number? (1) p and p
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Target question: Is p a prime number?ziyuenlau wrote:If p is a positive integer, is p a prime number?
(1) p and p+1 have the same number of factors.
(2) p−1 is a factor of p.
Given: p is a positive integer
Statement 1: p and p+1 have the same number of factors.
Let's TEST some values.
There are several values of p that satisfy statement 1. Here are two:
Case a: p = 2. This means that p+1 = 3. Notice that 2 and 3 both have the same number of factors (2 factors each). In this case, p IS prime
Case b: p = 14. This means that p+1 = 15. Notice that 14 and 15 both have the same number of factors (4 factors each). In this case, p is NOT prime
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: p−1 is a factor of p
Let's test some cases:
If p = 3, then p-1 = 2. Is 2 a factor of 3? No.
If p = 4, then p-1 = 3. Is 3 a factor of 4? No.
If p = 5, then p-1 = 4. Is 4 a factor of 5? No.
If p = 6, then p-1 = 5. Is 5 a factor of 6? No.
.
.
.
We can see that, if we keep going, p-1 will NEVER be a factor of p. Yet, statement 2 says that p-1 IS a factor of p.
Let's test the two positive integers that we haven't yet tested: 2 and 1
If p = 2, then p-1 = 1. Is 1 a factor of 2? YES! So, p COULD equal 2
If p = 1, then p-1 = 0. Is 0 a factor of 1? No
So, We can conclude that p MUST equal 2, which means p IS prime
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer: B
Cheers,
Brent
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Hi ziyuenlau/hazelnut01,
You posted this question this day before here:
https://www.beatthegmat.com/if-p-is-a-po ... 95194.html
This question can be solved by TESTing VALUES, but it will likely also require a bit of 'brute force' work. Sometimes the easiest/fastest way to get to the solution is to just put the pen on the pad and quickly list out the possibilities.
We're told that P is a POSITIVE INTEGER. We're asked if P is PRIME. This is a YES/NO question.
1) P and (P+1) have the same number of factors.
The information in this Fact might take a little bit of work to deal with, so let's brute force the possibilities until we find a couple of examples that match what we're told here:
P=1 .. 1 factor
P=2 .. 2 factors
P=3 .. 2 factors
P=4 .. 3 factors
P=5 .. 2 factors
P=6 .. 4 factors
P=7 .. 2 factors
P=8 .. 4 factors
P=9 .. 3 factors
P=10 .. 4 factors
P=11 .. 2 factors
P=12 .. 6 factors
P=13 .. 2 factors
P=14 .. 4 factors
P=15 .. 4 factors
We can now see two 'pairs' of numbers that have the SAME number of factors...
2 and 3; if P=2 then the answer to the question is YES
14 and 15; if P=14 then the answer to the question is NO
Fact 1 is INSUFFICIENT
2) (P-1) is a factor of P.
Again, let's start at P=1 and see what occurs...
P=1 .. 0 is not a factor of 1
P=2 .. 1 IS a factor of 2
P=3 .. 2 is not a factor of 3
P=4 .. 3 is not a factor of 4
Etc.
At this point, we can stop working - larger values of P will continue to yield the same result. The ONLY time that (P-1) is a factor of P is when P=2. Thus, there is ONLY one answer to the question (and it happens to be YES).
Fact 2 is SUFFICIENT
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
You posted this question this day before here:
https://www.beatthegmat.com/if-p-is-a-po ... 95194.html
This question can be solved by TESTing VALUES, but it will likely also require a bit of 'brute force' work. Sometimes the easiest/fastest way to get to the solution is to just put the pen on the pad and quickly list out the possibilities.
We're told that P is a POSITIVE INTEGER. We're asked if P is PRIME. This is a YES/NO question.
1) P and (P+1) have the same number of factors.
The information in this Fact might take a little bit of work to deal with, so let's brute force the possibilities until we find a couple of examples that match what we're told here:
P=1 .. 1 factor
P=2 .. 2 factors
P=3 .. 2 factors
P=4 .. 3 factors
P=5 .. 2 factors
P=6 .. 4 factors
P=7 .. 2 factors
P=8 .. 4 factors
P=9 .. 3 factors
P=10 .. 4 factors
P=11 .. 2 factors
P=12 .. 6 factors
P=13 .. 2 factors
P=14 .. 4 factors
P=15 .. 4 factors
We can now see two 'pairs' of numbers that have the SAME number of factors...
2 and 3; if P=2 then the answer to the question is YES
14 and 15; if P=14 then the answer to the question is NO
Fact 1 is INSUFFICIENT
2) (P-1) is a factor of P.
Again, let's start at P=1 and see what occurs...
P=1 .. 0 is not a factor of 1
P=2 .. 1 IS a factor of 2
P=3 .. 2 is not a factor of 3
P=4 .. 3 is not a factor of 4
Etc.
At this point, we can stop working - larger values of P will continue to yield the same result. The ONLY time that (P-1) is a factor of P is when P=2. Thus, there is ONLY one answer to the question (and it happens to be YES).
Fact 2 is SUFFICIENT
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
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Every integer is a factor of 0 except 0 itself.hoppycat wrote:This is kind of related. Is 1 a factor of 0? And is 0 a factor of 0?
Say m and n are integers. If m / n = an integer, then n is a factor of m.
0 / anything other than 0 = 0, an integer. It follows that anything other than 0 is a factor of 0.
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Other GMAT-relevant properties of 0:
* It's even
* It's neither positive nor negative
* It should ALWAYS be considered when a problem uses the phrase "nonpositive" or "nonnegative"
* 0! = 1
* 0 is a multiple of every integer
* 0 is not a factor of any integer
* If x ≠ 0, then x� = 1
* It's even
* It's neither positive nor negative
* It should ALWAYS be considered when a problem uses the phrase "nonpositive" or "nonnegative"
* 0! = 1
* 0 is a multiple of every integer
* 0 is not a factor of any integer
* If x ≠ 0, then x� = 1
Thanks!!Matt@VeritasPrep wrote:Other GMAT-relevant properties of 0:
* It's even
* It's neither positive nor negative
* It should ALWAYS be considered when a problem uses the phrase "nonpositive" or "nonnegative"
* 0! = 1
* 0 is a multiple of every integer
* 0 is not a factor of any integer
* If x ≠ 0, then x� = 1