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Fresh Meat!!! [#permalink] New post 17 Apr 2013, 05:11
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The next set of PS questions. I'll post OA's with detailed explanations after some discussion. Please, post your solutions along with the answers.

1. The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?

I. 63
II. 126
III. 252


A. I only
B. II only
C. III only
D. I and III only
E. I, II and III

Solution: fresh-meat-151046-80.html#p1215318


2. Set S contains 7 different letters. How many subsets of set S, including an empty set, contain at most 3 letters?

A. 29
B. 56
C. 57
D. 63
E. 64

Solution: fresh-meat-151046-100.html#p1215323

3. How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

A. 16
B. 27
C. 31
D. 32
E. 64

Solution: fresh-meat-151046-100.html#p1215329


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

Solution: fresh-meat-151046-100.html#p1215335


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

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

Solution: fresh-meat-151046-100.html#p1215338


6. If the least common multiple of a positive integer x, 4^3 and 6^5 is 6^6. Then x can take how many values?

A. 1
B. 6
C. 7
D. 30
E. 36

Solution: fresh-meat-151046-100.html#p1215345


7. The greatest common divisor of two positive integers is 25. If the sum of the integers is 350, then how many such pairs are possible?

A. 1
B. 2
C. 3
D. 4
E. 5

Solution: fresh-meat-151046-100.html#p1215349


8. The product of a positive integer x and 377,910 is divisible by 3,300, then the least value of x is:

A. 10
B. 11
C. 55
D. 110
E. 330

Solution: fresh-meat-151046-100.html#p1215359


9. What is the 101st digit after the decimal point in the decimal representation of 1/3 + 1/9 + 1/27 + 1/37?

A. 0
B. 1
C. 5
D. 7
E. 8

Solution: fresh-meat-151046-100.html#p1215367


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

Solution: fresh-meat-151046-100.html#p1215370


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Re: Fresh Meat!!! [#permalink] New post 21 Apr 2013, 23:07
Expert's post
Kudos points given for each correct solution.

Note that I cannot award more than 5 Kudos to the same person per day, so those of you who have more than 5 correct solutions please PM me tomorrow the links for which I owe you kudos points.

Thank you.

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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Fresh Meat!!! [#permalink] New post 22 Apr 2013, 19:24
This is a good collection of questions.
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Re: Fresh Meat!!! [#permalink] New post 25 Apr 2013, 18:42
Hi Bunuel,
Had the question been
Which of the following is a factor of 18!+1?

A. 15
B. 17
C. 19
D. 33
E. None of These

Then, would it be possible to come at a conclusion that 19 will be the factor of 18!+1.
In the original question, we came to the answer by eliminating other choices.

Please share your reasoning.

Thanks
H
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Re: Fresh Meat!!! [#permalink] New post 25 Apr 2013, 23:43
Expert's post
imhimanshu wrote:
Hi Bunuel,
Had the question been
Which of the following is a factor of 18!+1?

A. 15
B. 17
C. 19
D. 33
E. None of These

Then, would it be possible to come at a conclusion that 19 will be the factor of 18!+1.
In the original question, we came to the answer by eliminating other choices.

Please share your reasoning.

Thanks
H


Yes, we could get the correct answer with Wilson's theorem, but you don't need it for the GMAT.
_________________

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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Fresh Meat!!! [#permalink] New post 29 May 2013, 04:59
1 3 +1 9 +1 27 +1 37 =333 999 +111 999 +37 999 +27 999 =508 999 =0.508508... .

would you please explain how 1/37 is 27/999
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Re: Fresh Meat!!! [#permalink] New post 30 May 2013, 05:23
2. Set S contains 7 different letters. How many subsets of set S, including an empty set, contain at most 3 letters?

A. 29
B. 56
C. 57
D. 63
E. 64

Solution:

selecting 0 or more from n number of things is nC0+nC1+nC2+...+nCn
=> here it is selecting at most 3 letters so from 0 to 3 it is 7C0+7C1+7C2+7C3 = 64

Ans: E
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Re: Fresh Meat!!! [#permalink] New post 30 May 2013, 05:32
3. How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

A. 16
B. 27
C. 31
D. 32
E. 64

Solution:

Subset means selecting 0 or more from given set. since 0 should be excluded we have 5 numbers in set.

selecting 0 to n from given set is 2^n => 2^5= 32.

Ans: 32
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Re: Fresh Meat!!! [#permalink] New post 30 May 2013, 08:09
imhimanshu wrote:
Hi Bunuel,
Had the question been
Which of the following is a factor of 18!+1?

A. 15
B. 17
C. 19
D. 33
E. None of These

Then, would it be possible to come at a conclusion that 19 will be the factor of 18!+1.
In the original question, we came to the answer by eliminating other choices.

Please share your reasoning.

Thanks
H

Hi himanshu.
According to Wilson's Theorem, if P is a prime no. then the remainder when (p-1)! is divided by p is (p-1)
Therefore, 18! on division by 19 will give 18 as a remainder. Now 18+1 is divisible by 19 therefore answer to your query is 19.
add kudos if this helped you
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Re: Fresh Meat!!! [#permalink] New post 21 Jun 2013, 20:56
Hi Bunuel,
this seemed like a great way to earn kudos points,
Would you be having more questionaires like this in future also? as it sorta helps to boost up the kudos for people who have recently joined the forum and want to make it in time to get to the gmatclub tests by earning kudos.
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PS: Like my approach? Please Help me with some Kudos. :-)

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Re: Fresh Meat!!! [#permalink] New post 01 Jul 2013, 09:12
Hello.. Can someone please explain why "x" ( 15x:11x:9x) needs to be an integer? Why not 1.5?
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Re: Fresh Meat!!! [#permalink] New post 01 Jul 2013, 09:23
Expert's post
CIyer wrote:
Hello.. Can someone please explain why "x" ( 15x:11x:9x) needs to be an integer? Why not 1.5?


We are told that the length of the diagonals are integers and their ratio is 15:11:9. This means that the lengths are multiples of 15, 11 and 9. If x=1.5, then the lengths won't be integers.

Hope it's clear.
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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Fresh Meat!!! [#permalink] New post 07 Jul 2013, 06:56
Bunuel wrote:
3. How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

A. 16
B. 27
C. 31
D. 32
E. 64

Consider the set without 0: {1, 2, 3, 4, 5}. Each out of 5 elements of the set {1, 2, 3, 4, 5} has TWO options: either to be included in the subset or not, so total number of subsets of this set is 2^5=32. Now, each such set will be a subset of {0, 1, 2, 3, 4, 5} and won't include 0.

Answer: D.



Hi Bunuel,

I did this exercise as follows:

I eliminate the 0, so i have the following set: (1,2,3,4,5). Now, i use combinatorics.

Set containing 5 elements: 5C5=1
Set containing 4 elements: 4C5=5
Set containing 3 elements: 3C5=10
Set containing 2 elements: 2C5=10
Set containing 1 elements: 1C5=5

So, the total of posibilites are 31. What am I missing here¿??

Thanks in advance
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Re: Fresh Meat!!! [#permalink] New post 07 Jul 2013, 07:03
Expert's post
jacg20 wrote:
Bunuel wrote:
3. How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

A. 16
B. 27
C. 31
D. 32
E. 64

Consider the set without 0: {1, 2, 3, 4, 5}. Each out of 5 elements of the set {1, 2, 3, 4, 5} has TWO options: either to be included in the subset or not, so total number of subsets of this set is 2^5=32. Now, each such set will be a subset of {0, 1, 2, 3, 4, 5} and won't include 0.

Answer: D.



Hi Bunuel,

I did this exercise as follows:

I eliminate the 0, so i have the following set: (1,2,3,4,5). Now, i use combinatorics.

Set containing 5 elements: 5C5=1
Set containing 4 elements: 4C5=5
Set containing 3 elements: 3C5=10
Set containing 2 elements: 2C5=10
Set containing 1 elements: 1C5=5

So, the total of posibilites are 31. What am I missing here¿??

Thanks in advance


You are missing 1 empty set, which is a subset of the original set and also does not contain 0.

Hope it's clear.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Fresh Meat!!! [#permalink] New post 07 Jul 2013, 07:06
Bunuel wrote:
jacg20 wrote:
Bunuel wrote:
3. How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

A. 16
B. 27
C. 31
D. 32
E. 64

Consider the set without 0: {1, 2, 3, 4, 5}. Each out of 5 elements of the set {1, 2, 3, 4, 5} has TWO options: either to be included in the subset or not, so total number of subsets of this set is 2^5=32. Now, each such set will be a subset of {0, 1, 2, 3, 4, 5} and won't include 0.

Answer: D.



Hi Bunuel,

I did this exercise as follows:

I eliminate the 0, so i have the following set: (1,2,3,4,5). Now, i use combinatorics.

Set containing 5 elements: 5C5=1
Set containing 4 elements: 4C5=5
Set containing 3 elements: 3C5=10
Set containing 2 elements: 2C5=10
Set containing 1 elements: 1C5=5

So, the total of posibilites are 31. What am I missing here¿??

Thanks in advance


You are missing 1 empty set, which is a subset of the original set and also does not contain 0.

Hope it's clear.


Ouch, tricky.. Thanks so much!
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Re: Fresh Meat!!! [#permalink] New post 08 Jul 2013, 08:03
1 useful formula: 2^n=nc0 + nc1+ ..... + ncn; here n=5, ans= 2^5= 32
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Re: Fresh Meat!!! [#permalink] New post 08 Jul 2013, 09:46
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Re: Fresh Meat!!! [#permalink] New post 08 Jul 2013, 10:26
Bunuel wrote:
SOLUTIONS:

1. The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?

I. 63
II. 126
III. 252


A. I only
B. II only
C. III only
D. I and III only
E. I, II and III

Given that the ratio of the diagonal is d_s:d_1:d_2=15x:11x:9x, for some positive integer x (where d_s is the diagonal of square S and d_1 and d_2 are the diagonals of rhombus R).

area_{square}=\frac{d^2}{2} and area_{rhombus}=\frac{d_1*d_2}{2}.

The difference is area_{square}-area_{rhombus}=\frac{(15x)^2}{2}-\frac{11x*9x}{2}=63x^2.

If x=1, then the difference is 63;
If x=2, then the difference is 252;
In order the difference to be 126 x should be \sqrt{2}, which is not possible.

Answer: D.


Hi Bunuel,

This is probably a stupid question. But why can't x be \sqrt{2}?
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Re: Fresh Meat!!! [#permalink] New post 08 Jul 2013, 10:28
Aho92 wrote:
Bunuel wrote:
SOLUTIONS:

1. The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?

I. 63
II. 126
III. 252


A. I only
B. II only
C. III only
D. I and III only
E. I, II and III

Given that the ratio of the diagonal is d_s:d_1:d_2=15x:11x:9x, for some positive integer x (where d_s is the diagonal of square S and d_1 and d_2 are the diagonals of rhombus R).

area_{square}=\frac{d^2}{2} and area_{rhombus}=\frac{d_1*d_2}{2}.

The difference is area_{square}-area_{rhombus}=\frac{(15x)^2}{2}-\frac{11x*9x}{2}=63x^2.

If x=1, then the difference is 63;
If x=2, then the difference is 252;
In order the difference to be 126 x should be \sqrt{2}, which is not possible.

Answer: D.


Hi Bunuel,

This is probably a stupid question. But why can't x be \sqrt{2}?

Okay. I got it. Stupid me. They have to be integers
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Re: Fresh Meat!!! [#permalink] New post 08 Jul 2013, 23:40
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Re: Fresh Meat!!! [#permalink] New post 22 Jul 2013, 08:08
hi Bunuel, thanks so much for your help - you're explanations to all kinds of problems have been invaluable to me in my studies thus far...

RE: the following question -

3. How many different subsets of the set {0, 1, 2, 3, 4, 5} do not contain 0?

A. 16
B. 27
C. 31
D. 32
E. 64

Solution:

Subset means selecting 0 or more from given set. since 0 should be excluded we have 5 numbers in set.

selecting 0 to n from given set is 2^n => 2^5= 32.

Ans: 32


Where do you get the 2 rom in 2*5?
Re: Fresh Meat!!!   [#permalink] 22 Jul 2013, 08:08
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