M25-28

Official Solution:

The functions \(f\) and \(g\) are defined for all positive integers \(n\) as follows: \(f(n)\) represents the number of positive perfect squares less than \(n\), and \(g(n)\) represents the number of prime numbers less than \(n\). If \(f(x) + g(x) = 16\), then \(x\) is in the range:

A. \(29 \lt x \lt 35\)
B. \(30 \lt x \lt 36\)
C. \(31 \lt x \lt 37\)
D. \(32 \lt x \lt 38\)
E. \(33 \lt x \lt 39\)


Positive perfect squares: 1, 4, 9, 16, 25, 36, ...

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...

If \(x = 31\), then \(f(31) = 5\) and \(g(31) = 10\): \(f(x) + g(x) = 5 + 10 = 15\).

If \(x = 32\), then \(f(32) = 5\) and \(g(32) = 11\): \(f(x) + g(x) = 5 + 11 = 16\).

...

If \(x = 36\), then \(f(36) = 5\) and \(g(36) = 11\): \(f(x) + g(x) = 5 + 11 = 16\).

If \(x = 37\), then \(f(37) = 6\) and \(g(37) = 11\): \(f(x) + g(x) = 6 + 11 = 17\).

Thus, the possible values for \(x\) are 32, 33, 34, 35, or 36, which means \(x\) is in the range \(31 \lt x \lt 37\).


Answer: C