swerve wrote: ↑

Wed Mar 25, 2020 11:59 am

Let @x@ be defined as the number of positive perfect squares less than x
Let #x# be defined as the number of primes less than @x@
If #x# = 5, what is the value of #(@x@)#?

A. 7
B. 5
C. 4
D. 2
E. 1

The OA is E

Source: Magoosh

Since #x# = 5, we see that there are 5 primes less than @x@. Since the 5th prime is 11 and the 6th prime is 13, we see that @x@ must be 12 or 13. Therefore, we have:

#(@x@)# = #12# or #13#

Since #12# is the number of primes less than @12@ and @12@ is the number of positive perfect squares less than 12, @12@ = 3 (since there are 3 perfect squares less than 12: 1, 4 and 9). Notice that since @13@ = 3 also, #13# is equal to the number of primes less than 3 as well.

So, #12# = #13# is the number of primes less than 3, and since there is only 1 prime (namely, 2) less than 3, the answer is 1.

Answer: E