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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