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# Fresh Meat!!!

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Intern
Joined: 14 Jan 2017
Posts: 9

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30 May 2018, 04:31
Bunuel wrote:
4. 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.

It is given "f(n) is the number of positive perfect squares less than n" then X should be greater than 36, with this answer comes to D. ??? please correct if I am wrong & please correct.
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Joined: 02 Sep 2009
Posts: 49276

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30 May 2018, 08:55
thunderbird350 wrote:
Bunuel wrote:
4. 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.

It is given "f(n) is the number of positive perfect squares less than n" then X should be greater than 36, with this answer comes to D. ??? please correct if I am wrong & please correct.

It's not clear from your post as to why should x be greater than 36. The solution you quote explains that x could be 32, 33, 34, 35 or 36: 31<x<37, which is answer C.
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Joined: 24 Mar 2018
Posts: 91

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23 Jul 2018, 02:47
Bunuel wrote:
10. If x is not equal to 0 and x^y=1, then which of the following must be true?

I. x=1
II. x=1 and y=0
III. x=1 or y=0

A. I only
B. II only
C. III only
D. I and III only
E. None

Notice that if x=-1 and y is any even number, then $$(-1)^{even}=1$$, thus none of the options must be true.

Bunuel
If we consider only I
i.e x=1
1 to power anything will be one so shouldn't the OA be A ?
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Joined: 02 Sep 2009
Posts: 49276

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23 Jul 2018, 02:54
teaserbae wrote:
Bunuel wrote:
10. If x is not equal to 0 and x^y=1, then which of the following must be true?

I. x=1
II. x=1 and y=0
III. x=1 or y=0

A. I only
B. II only
C. III only
D. I and III only
E. None

Notice that if x=-1 and y is any even number, then $$(-1)^{even}=1$$, thus none of the options must be true.

Bunuel
If we consider only I
i.e x=1
1 to power anything will be one so shouldn't the OA be A ?

The question asks: which of the following MUST be true, not COULD be true. Now, ask yourself is x = 1 ALWAYS true? Doesn't the solution you quote, gives an example when x = 1 is NOT true?
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Joined: 24 Mar 2018
Posts: 91

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Updated on: 23 Jul 2018, 03:10
Bunuel wrote:
teaserbae wrote:
Bunuel wrote:
10. If x is not equal to 0 and x^y=1, then which of the following must be true?

I. x=1
II. x=1 and y=0
III. x=1 or y=0

A. I only
B. II only
C. III only
D. I and III only
E. None

Notice that if x=-1 and y is any even number, then $$(-1)^{even}=1$$, thus none of the options must be true.

Bunuel
If we consider only I
i.e x=1
1 to power anything will be one so shouldn't the OA be A ?

The question asks: which of the following MUST be true, not COULD be true. Now, ask yourself is x = 1 ALWAYS true? Doesn't the solution you quote, gives an example when x = 1 is NOT true?

Bunuel
Yeah I got that but what's wrong with C ?
X=1 than x^y=1
y=0 than x^y=1
there doesn't exsist any other case for which x^y is not equal to 1

Originally posted by teaserbae on 23 Jul 2018, 02:57.
Last edited by teaserbae on 23 Jul 2018, 03:10, edited 2 times in total.
Math Expert
Joined: 02 Sep 2009
Posts: 49276

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23 Jul 2018, 03:09
teaserbae wrote:
Bunuel wrote:
teaserbae wrote:
10. If x is not equal to 0 and x^y=1, then which of the following must be true?

I. x=1
II. x=1 and y=0
III. x=1 or y=0

A. I only
B. II only
C. III only
D. I and III only
E. None

Bunuel
If we consider only I
i.e x=1
1 to power anything will be one so shouldn't the OA be A ?

The question asks: which of the following MUST be true, not COULD be true. Now, ask yourself is x = 1 ALWAYS true? Doesn't the solution you quote, gives an example when x = 1 is NOT true?

Yeah I got that but what's wrong with C ?
X=1 than x^y=1
y=0 than x^y=1
there doesn't exsist any other case for which x^y is not equal to 1

I'll copy my solution here:

Notice that if x=-1 and y is any even number, then $$(-1)^{even}=1$$, thus none of the options must be true.

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Joined: 24 Mar 2018
Posts: 91

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23 Jul 2018, 03:16
Bunuel wrote:
teaserbae wrote:
Bunuel wrote:
10. If x is not equal to 0 and x^y=1, then which of the following must be true?

I. x=1
II. x=1 and y=0
III. x=1 or y=0

A. I only
B. II only
C. III only
D. I and III only
E. None

Bunuel
If we consider only I
i.e x=1
1 to power anything will be one so shouldn't the OA be A ?

The question asks: which of the following MUST be true, not COULD be true. Now, ask yourself is x = 1 ALWAYS true? Doesn't the solution you quote, gives an example when x = 1 is NOT true?

I'll copy my solution here:

Notice that if x=-1 and y is any even number, then $$(-1)^{even}=1$$, thus none of the options must be true.

Bunuel
By III x=1 or y=0
since x= -1 so y=0
Isn't (-1)^0 = 1 ?
As 0 is even integer
Math Expert
Joined: 02 Sep 2009
Posts: 49276

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23 Jul 2018, 03:24
1
teaserbae wrote:
Bunuel wrote:
teaserbae wrote:
10. If x is not equal to 0 and x^y=1, then which of the following must be true?

I. x=1
II. x=1 and y=0
III. x=1 or y=0

A. I only
B. II only
C. III only
D. I and III only
E. None

The question asks: which of the following MUST be true, not COULD be true. Now, ask yourself is x = 1 ALWAYS true? Doesn't the solution you quote, gives an example when x = 1 is NOT true?

I'll copy my solution here:

Notice that if x=-1 and y is any even number, then $$(-1)^{even}=1$$, thus none of the options must be true.

Bunuel
By III x=1 or y=0
since x= -1 so y=0
Isn't (-1)^0 = 1 ?
As 0 is even integer

I'll try this last time. The question asks: which of the following MUST be true?

None of the options is necessarily true because if x = -1, and say y = 2, then x^y = 1 and not I, not II and not III is true.
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Joined: 18 Jun 2018
Posts: 48
Location: United States
Concentration: Finance, Healthcare

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09 Sep 2018, 11:48
Hello Bunuel,

You mentioned that we can factor 17 in the solution to this problem, is that a typo?

Thanks.

Bunuel wrote:
5. Which of the following is a factor of 18!+1?

A. 15
B. 17
C. 19
D. 33
E. 39

18! and 18!+1 are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1.

Now, since we can factor out each 15, 17, 33=3*11, and 39=3*13 out of 18!, then 15, 17, 33 and 39 ARE factors of 18! and are NOT factors of 18!+1. Therefore only 19 could be a factor of 18!+1.

Math Expert
Joined: 02 Sep 2009
Posts: 49276

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09 Sep 2018, 20:38
1
funsogu wrote:
Hello Bunuel,

You mentioned that we can factor 17 in the solution to this problem, is that a typo?

Thanks.

Bunuel wrote:
5. Which of the following is a factor of 18!+1?

A. 15
B. 17
C. 19
D. 33
E. 39

18! and 18!+1 are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1.

Now, since we can factor out each 15, 17, 33=3*11, and 39=3*13 out of 18!, then 15, 17, 33 and 39 ARE factors of 18! and are NOT factors of 18!+1. Therefore only 19 could be a factor of 18!+1.

17 is a factor of 18!:

18! = 1*2*3*...*17*18.
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Re: Fresh Meat!!! &nbs [#permalink] 09 Sep 2018, 20:38

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