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Intern  Status: Learning
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The functions f and g are defined for all the positive  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 36% (02:17) correct 64% (02:51) wrong based on 123 sessions

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The functions f and g are defined for all the positive integers n by the following rule: f(n) is the number of perfect squares less than n and g(n) is the number of primes numbers less than n. If f(x) + g(x) = 16, then x is in the range:

A. 30 < x < 36

B. 30 < x < 37

C. 31 < x < 37

D. 31 < x < 38

E. 32 < x < 38

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GMAT 1: 730 Q52 V37 Re: The functions f and g are defined for all the positive  [#permalink]

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5
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.
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Re: The functions f and g are defined for all the positive  [#permalink]

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farful wrote:
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.

Than why not not f(x)=6 and g(x)=10?
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GMAT 1: 730 Q52 V37 Re: The functions f and g are defined for all the positive  [#permalink]

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ronr34 wrote:
farful wrote:
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.

Than why not not f(x)=6 and g(x)=10?

f(x) >= 6 implies x > 36.
g(x) <= 10 implies x < 32.
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Re: The functions f and g are defined for all the positive  [#permalink]

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The functions f and g are defined for all the positive integers n by the following rule: f(n) is the number of positive perfect squares less than n and g(n) is the number of primes numbers less than n. If f(x) + g(x) = 16, then x is in the range:

A. 30 < x < 36
B. 30 < x < 37
C. 31 < x < 37
D. 31 < x < 38
E. 32 < x < 38

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 x could be 32, 33, 34, 35 or 36: 31<x<37.

P.S. This is GMAT Club's question, not Grockit's.
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: The functions f and g are defined for all the positive  [#permalink]

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Hi All,

We can use the answer choices to our advantage on this question (by determining if number of possible or impossible, we can eliminate answer choices).

Let's start by TESTing X=31. According to the prompt...

f(31) = number of perfect squares less than 31 = {1, 4, 9, 16, 25] = 5
g(31) = number of primes less than 31 = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29} = 10

f(31) + g(31) = 15; we're told that it's SUPPOSED to be 16 though, so X CANNOT be 31. Eliminate Answers A and B.

Next, let's TEST X = 37. According to the prompt...

f(37) = number of perfect squares less than 31 = {1, 4, 9, 16, 25, 36] = 6
g(37) = number of primes less than 31 = {2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 3} = 11

f(37) + g(37) = 17; we're told that it's SUPPOSED to be 16 though, so X CANNOT be 37. Eliminate Answers D and E.

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Rich
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Re: The functions f and g are defined for all the positive  [#permalink]

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farful wrote:
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.

Why are we not considering 0 as a perfect square?
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Re: The functions f and g are defined for all the positive  [#permalink]

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Bunuel wrote:
The functions f and g are defined for all the positive integers n by the following rule: f(n) is the number of positive perfect squares less than n and g(n) is the number of primes numbers less than n. If f(x) + g(x) = 16, then x is in the range:

A. 30 < x < 36
B. 30 < x < 37
C. 31 < x < 37
D. 31 < x < 38
E. 32 < x < 38

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 x could be 32, 33, 34, 35 or 36: 31<x<37.

P.S. This is GMAT Club's question, not Grockit's.

Why are we not considering 0 as a perfect square?
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Joined: 02 Sep 2009
Posts: 58453
Re: The functions f and g are defined for all the positive  [#permalink]

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1
shwetakoshija wrote:
Bunuel wrote:
The functions f and g are defined for all the positive integers n by the following rule: f(n) is the number of positive perfect squares less than n and g(n) is the number of primes numbers less than n. If f(x) + g(x) = 16, then x is in the range:

A. 30 < x < 36
B. 30 < x < 37
C. 31 < x < 37
D. 31 < x < 38
E. 32 < x < 38

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 x could be 32, 33, 34, 35 or 36: 31<x<37.

P.S. This is GMAT Club's question, not Grockit's.

Why are we not considering 0 as a perfect square?

Check the highlighted parts.
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Re: The functions f and g are defined for all the positive  [#permalink]

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