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09 May 2014, 06:00
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39% (02:29) correct
61%
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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. \(33 \lt x \lt 39\)
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. \(33 \lt x \lt 39\)
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09 May 2014, 06:58
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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
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
General Discussion
09 May 2014, 06:26
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f(n) = number of perfect squares less than n
1,4,9,16,25,36
f(25) = 4
f(26)=f(27)=...=f(35)=f(36) = 5
f(37) = 6
g(n) = number of prime numbers less than n
2,3,5,7,11,13,17,19,23,29,31,37
g(31)=10
g(32)=g(33)=...=g(36)=g(37) = 11
g(38)=12
f(x)+g(x)=16 implies that f(x)=5 and g(x)=11.
So, x = 32,33,34,35,or36.
Answer C, 31 < x < 37.
1,4,9,16,25,36
f(25) = 4
f(26)=f(27)=...=f(35)=f(36) = 5
f(37) = 6
g(n) = number of prime numbers less than n
2,3,5,7,11,13,17,19,23,29,31,37
g(31)=10
g(32)=g(33)=...=g(36)=g(37) = 11
g(38)=12
f(x)+g(x)=16 implies that f(x)=5 and g(x)=11.
So, x = 32,33,34,35,or36.
Answer C, 31 < x < 37.
