Problem of the Day


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

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

(2) 20\leq a <29

Reveal Answer

Answer

C. Both statements together are sufficient, but neither statement alone is sufficient.

See the Solution

Solution

[latexpage]

First let’s explore how $T(n)$ works with a few examples. $T(4)=2$ because there are two prime numbers less than 4: 2 and 3. $T(9)=4$ because there are four prime numbers less than 9: 2, 3, 5, 7.

Statement 1: $\displaystyle \frac{T(a+1)}{T(a)}-1= \frac{1}{T(a)}$. We can rewrite this by multiplying both sides  of the equation by $T(a)$ to get $T(a+1)-T(a)=1$. This implies that $a$ is prime, but it does not tell us the exact value of $a$. INSUFFICIENT.

Statement 2: $20 \leq a <29$. This is clearly insufficient, as it only tells us a range of possible values for $a$. INSUFFICIENT.

Statements 1&2: Combining the statements we know that $a$ is prime, and $20 \leq a <29$ The only possibility is $a=23$. SUFFICIENT.


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