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Bunuel
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Let T(n) represent the number of prime numbers less than n. What is th 03 Dec 2019, 01:28
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C
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61% (02:28) correct 39% (02:20) wrong based on 201 sessions

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Let T(n) represent the number of prime numbers less than n. What is the value of positive integer a?

(1) \(\frac{T(a+1)}{T(a)}-1=\frac{1}{T(a)}\)

(2) \(20\leq a <29\)


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C
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Let T(n) represent the number of prime numbers less than n. What is th 03 Dec 2019, 02:20
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This is a very good question on the understanding of prime number.
the solution is as attached.


Bunuel
Let T(n) represent the number of prime numbers less than n. What is the value of positive integer a?

(1) \(\frac{T(a+1)}{T(a)}-1=\frac{1}{T(a)}\)

(2) \(20\leq a <29\)


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WhatsApp Image 2019-12-03 at 2.47.16 PM.jpeg [ 57.03 KiB | Viewed 3901 times ]
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Let T(n) represent the number of prime numbers less than n. What is th 03 Dec 2019, 02:58
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gmatbusters
This is a very good question on the understanding of prime number.
the solution is as attached.


Bunuel
Let T(n) represent the number of prime numbers less than n. What is the value of positive integer a?

(1) \(\frac{T(a+1)}{T(a)}-1=\frac{1}{T(a)}\)

(2) \(20\leq a <29\)


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T(21)=T(20+1)=21-8=13
T(20)=20-8=12
a=20
again
T(22)=22-8=14
T(21)=21-8=13
a=21
why not E? Could you please where I am going wrong?
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Let T(n) represent the number of prime numbers less than n. What is th 03 Dec 2019, 03:17
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you interpreted the meaning of T(n) incorrectly.
T(20) = number of prime numbers less than 20
prime numbers less than 20 are 2,3,5,7,11,13,17,29.
Hence T(20) = 8
similarly for other T(21) = 8 only

satya2029
gmatbusters
This is a very good question on the understanding of prime number.
the solution is as attached.


Bunuel
Let T(n) represent the number of prime numbers less than n. What is the value of positive integer a?

(1) \(\frac{T(a+1)}{T(a)}-1=\frac{1}{T(a)}\)

(2) \(20\leq a <29\)


Are You Up For the Challenge: 700 Level Questions
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Attachment:
WhatsApp Image 2019-12-03 at 2.47.16 PM.jpeg

T(21)=T(20+1)=21-8=13
T(20)=20-8=12
a=20
again
T(22)=22-8=14
T(21)=21-8=13
a=21
why not E? Could you please where I am going wrong?
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Let T(n) represent the number of prime numbers less than n. What is th 03 Dec 2019, 04:10
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satya2029
gmatbusters
This is a very good question on the understanding of prime number.
the solution is as attached.


Bunuel
Let T(n) represent the number of prime numbers less than n. What is the value of positive integer a?

(1) \(\frac{T(a+1)}{T(a)}-1=\frac{1}{T(a)}\)

(2) \(20\leq a <29\)


Are You Up For the Challenge: 700 Level Questions
Show Spoiler
Attachment:
WhatsApp Image 2019-12-03 at 2.47.16 PM.jpeg

T(21)=T(20+1)=21-8=13
T(20)=20-8=12
a=20
again
T(22)=22-8=14
T(21)=21-8=13
a=21
why not E? Could you please where I am going wrong?
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Thanks:)
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Let T(n) represent the number of prime numbers less than n. What is th 05 Dec 2019, 04:29
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Bunuel
Let T(n) represent the number of prime numbers less than n. What is the value of positive integer a?

(1) \(\frac{T(a+1)}{T(a)}-1=\frac{1}{T(a)}\)

(2) \(20\leq a <29\)
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(1) \(\frac{T(a+1)}{T(a)}-1=\frac{1}{T(a)}\) insufic

\(\frac{T(a+1)}{T(a)}-1=\frac{1}{T(a)}…T(a+1)-T(a)=1\)

\(T(a,a+1)=T(3,4)…T(a)=[2]=1…T(a+1)=[2,3]=2…T(a+1)-T(a)=1=TRUE\)
\(T(a,a+1)=T(5,6)…T(a)=[2,3]=2…T(a+1)=[2,3,5]=3…T(a+1)-T(a)=1=TRUE\)

Any \(a = prime\) fits the description.

(2) \(20\leq a <29\) insufic

(1)&(2) sufic

\(20\leq a <29…only.prime=23…a=23\)

Ans (C)
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Let T(n) represent the number of prime numbers less than n. What is th 05 Dec 2019, 07:05
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Bunuel
Let T(n) represent the number of prime numbers less than n. What is the value of positive integer a?

(1) \(\frac{T(a+1)}{T(a)}-1=\frac{1}{T(a)}\)

(2) \(20\leq a <29\)


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Analyzing the question:
It may help to visual what T(n) represents. If n = 10, the primes less than 10 are 2, 3, 5, 7. So T(10) = 4. If n = 11, it is the same list so T(11) = 4. However, for n = 12, we can include 11 in the prime list so T(12) = 5.

Statement 1:

T(n) in a fraction form doesn't really make sense so let's try to get rid of the fraction by multiplying both sides by T(a).

We get: \(T(a + 1) - T(a) = 1 \) or \(T(a + 1) = T(a) + 1\)
Now T(a) represents the amount of primes less than \(a\). So from \(a\) to a + 1, T(a + 1) must have included another prime in the list, in order to have T(a + 1) just one bigger than T(a).
Hence \(a\) must be prime. Now finally we simplified the statement to: \(a\) is prime. Of course, there are many primes so this is insufficient.

Statement 2:
Insufficient.

Combined:
The only prime in the range is a = 23, Sufficient.

Ans: C
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Let T(n) represent the number of prime numbers less than n. What is th 05 Dec 2019, 19:22
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Shouldn't this question come with the condition that n >2,
for 1/T(n) becomes not defined in that case
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Let T(n) represent the number of prime numbers less than n. What is th 05 Dec 2019, 19:37
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Let T(n) represent the number of prime numbers less than n. What is the value of positive integer a?

(1) \(\frac{T(a+1)}{T(a)}-1=\frac{1}{T(a)}..........\frac{T(a+1)-T(a)}{T(a)}=\frac{1}{T(a)}......T(a+1)-T(a)=1\)
So, a+1 has ONE more prime number than a, meaning that a itself is a prime number. a can take infinite values, 3, 5, etc.

(2) \(20\leq a <29\)
a could be any of the values from 20 to 28

Combined..
There is ONLY one prime number 23 in range \(20\leq a <29\), so a=23

C

Yes, LJ11 it could have been written that a>2. But even if it is not written, the moment there is a statement, that is, it is given that 1/x=1/y is true, it should mean that x and y are not 0.
And if I ask you 'is 1/x=1/y', then yes it should surely be mentioned that x and y are not 0.
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Let T(n) represent the number of prime numbers less than n. What is th 13 May 2022, 17:50
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GMATBusters
This is a very good question on the understanding of prime number.
the solution is as attached.


Bunuel
Let T(n) represent the number of prime numbers less than n. What is the value of positive integer a?

(1) \(\frac{T(a+1)}{T(a)}-1=\frac{1}{T(a)}\)

(2) \(20\leq a <29\)


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What if a=28
Then a+1 = 29

then T(a+1) = T(a) + 1
Ie T(29) = T(28) + 1
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Let T(n) represent the number of prime numbers less than n. What is th 03 Jul 2024, 00:40
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