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Fresh Meat!!!

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Math Expert
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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

Kudos points for each correct solution!!!
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21 Apr 2013, 22:45
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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

$$\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}=\frac{333}{999} + \frac{111}{999} + \frac{37}{999} + \frac{27}{999}=\frac{508}{999}=0.508508...$$.

102nd digit will be 8, thus 101st digit will be 0.

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21 Apr 2013, 23:07
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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22 Apr 2013, 19:24
This is a good collection of questions.
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Fugitive

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25 Apr 2013, 18:42
Hi Bunuel,
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.

Thanks
H
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Math Expert
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Kudos [?]: 93656 [0], given: 10583

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25 Apr 2013, 23:43
imhimanshu wrote:
Hi Bunuel,
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.

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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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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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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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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30 May 2013, 08:09
imhimanshu wrote:
Hi Bunuel,
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.

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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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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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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01 Jul 2013, 09:23
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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07 Jul 2013, 07:03
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.

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¿??

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

Hope it's clear.
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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.

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¿??

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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08 Jul 2013, 08:03
1 useful formula: 2^n=nc0 + nc1+ ..... + ncn; here n=5, ans= 2^5= 32
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08 Jul 2013, 09:46
CIyer wrote:
1 useful formula: 2^n=nc0 + nc1+ ..... + ncn; here n=5, ans= 2^5= 32

That's correct. Another way of reaching the same formual is here: fresh-meat-151046-100.html#p1215329
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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.

Hi Bunuel,

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

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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08 Jul 2013, 23:40
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE
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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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