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Let T(n) represent the number of prime numbers less than n. What is th

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Let T(n) represent the number of prime numbers less than n. What is th  [#permalink]

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New post 03 Dec 2019, 01:28
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Re: Let T(n) represent the number of prime numbers less than n. What is th  [#permalink]

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New post 03 Dec 2019, 02:20
2
This is a very good question on the understanding of prime number.
the solution is as attached.
Image

Bunuel wrote:
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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Attachment:
WhatsApp Image 2019-12-03 at 2.47.16 PM.jpeg
WhatsApp Image 2019-12-03 at 2.47.16 PM.jpeg [ 57.03 KiB | Viewed 475 times ]

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Re: Let T(n) represent the number of prime numbers less than n. What is th  [#permalink]

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New post 03 Dec 2019, 02:58
gmatbusters wrote:
This is a very good question on the understanding of prime number.
the solution is as attached.
Image

Bunuel wrote:
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

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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Re: Let T(n) represent the number of prime numbers less than n. What is th  [#permalink]

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New post 03 Dec 2019, 03:17
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 wrote:
gmatbusters wrote:
This is a very good question on the understanding of prime number.
the solution is as attached.
Image

Bunuel wrote:
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

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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Re: Let T(n) represent the number of prime numbers less than n. What is th  [#permalink]

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New post 03 Dec 2019, 04:10
satya2029 wrote:
gmatbusters wrote:
This is a very good question on the understanding of prime number.
the solution is as attached.
Image

Bunuel wrote:
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

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?

Thanks:)
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Re: Let T(n) represent the number of prime numbers less than n. What is th  [#permalink]

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New post 05 Dec 2019, 04:29
Bunuel wrote:
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) \(20\leq a <29\)



(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  [#permalink]

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New post 05 Dec 2019, 07:05
Bunuel wrote:
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


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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Re: Let T(n) represent the number of prime numbers less than n. What is th  [#permalink]

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New post 05 Dec 2019, 19:22
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  [#permalink]

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New post 05 Dec 2019, 19:37
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   [#permalink] 05 Dec 2019, 19:37
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