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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 17 Apr 2013, 09:36
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Q6.

LCM of x, 2^6 and (2*3)^5 = (2*3)^6
therefore x must at least have a 3^6 in it. Plus it can have other powers of 2 ranging from 2^0 to 2^6
that makes 7 possible values for X
And C
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 09:43
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Q7,
gcf = 25 = 5^2
Sum of 2 integers = 350

25 , 325 -> GCF = 25, SUM = 350 (OK)
50 , 300 -> GCF = 50, SUM = 350 (NOT OK)
75 , 275 -> GCF = 25, SUM = 350 (OK)
100 , 250 -> GCF = 50, SUM = 350 (NOT OK)
125 , 225 -> GCF = 25, SUM = 350 (OK)
150 , 200 -> GCF = 50, SUM = 350 (NOT OK)
175 , 175 -> GCF = 175, SUM = 350 (NOT OK)

Hence there will be 3 such pairs of integers.
Ans C
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 09:48
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Q8

x * 2 * (3^2) * 5 * 13 * 17 * 19 / (2^2 * 3 * 5^2 * 11)
after simplification we are left with:

x * 3 * 13 * 17 * 19 / 2 * 5 * 11

therefore x must at least be divisible by 110 (2*5*11)

Ans D
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 09:54
Q9.
reducing the equation makes it 508/999
when any number is divided by 999, the remainder after the decimal keeps repeating itself infinetely.

there the above fraction will be 0.508508508 ... and so on
every 2, 5, 8th ... didgit will be 0 and
every 3, 6, 9, 12th ... didgit will be 8
since 101 = 3*x + 2
101st digit in decimal will be 0
ans A
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 09:55
Bunuel wrote:
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.

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

Kudos points for each correct solution!!!

-----------

If x is not equal to 0 and x^y=1, then which of the following must be true?

The Q itself says that x is not equal to zero. Therefore, x can be +ve or -ve both okay.
now, option 1 says...... x=1 it is false becoz it can't be must be true as x can be one but it can be some other number as well. for ex. this also satisfies the condition given. 2^0=1. therefore it must not be true.

Option second says ...... II. x=1 and y=0 , similar to the above one the given condition can be true but it is not
must to be true as 2^0=1 again satisfies the condition given. Therefore False for me again.
Option -III. it says either x=1 or y=0 now this option fulfills the criteria as if x =1 then no matter the power raise to it it will alwz remains 1 or if y will be equal to zero then no matter the base but the exp. remains in every case. Hence,
III must be true ...... Therefore, C.
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 10:12
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Bunuel wrote:
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.

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
Kudos points for each correct solution!!!

----

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

Now in this Q, the Q for at most 3 letters in a subset that means it can have 0 or 1 or 2 or 3.
Therefore, when the set contains 0, it can be represented as 7c0 which yields us 1.
Second.... when the set contains 1, it can be represented as 7c1 which yields us 7.
Third, .........when the set contains 2, it can be represented as 7c2 which yields us 21.
Forth, .........when the set contains 3, it can be represented as 7c3 which yields us 35.
Now as we know that these four conditions are joined together with the help of "or"
so we must add them.... Therefore, 1+7+21+35=64. Hence, 64 answer....... E.
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 10:25
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Bunuel wrote:
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.

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

A. 15
B. 17
C. 19
D. 33
E. 39
Kudos points for each correct solution!!!

------------------------------------------------------------------------------------------------------------

First of all, before solving this problem, we must remember that two consecutive integers have a GCF of 1. That means 2 consecutive integers only share one factor that is 1. They never share any other factor. therefore, the factors of 18! can never be the factors of 18!+1. Okay!!!

Now, lets take a look at the answer choices.
1. 15, it is a factor of 18! as 18! contains both 5 & 3. Therefore, Eliminated.
2. 17, it is a factor of 18! as 18! contains 17. Therefore, Eliminated.
3. 19, it is not a factor of 18! as 18! can't contain 19 in it. Therefore, Chosen... Correct.
4. 33, it is a factor of 18! as 18! contains both 11 & 3. Therefore, Eliminated.
5. 39, it is a factor of 18! as 18! contains both 13 & 3. Therefore, Eliminated.
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 10:37
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Bunuel wrote:
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.

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
Kudos points for each correct solution!!!

--------------------------------------------------------------------------------------------------------------

In this Q. we have to find the least value of X ????

Now for this first of all we have prime factorize 377,910 i.e.;

377,910 = 4199*3^3*2*5
3300 = 2*3*5*11*2*5.

Now as given in the Question that, The product of a positive integer x and 377,910 is divisible by 3,300.
Therefore, 4199*3^3*2*5*x / 2*3*5*11*2*5.

After cancelling the required. we get X= 2*5*11 =110.
Therefore 110........................ D.
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 10:48
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Bunuel wrote:
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.

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
Kudos points for each correct solution!!!

-------------------------------------------------------------------------------------------------------------------

Here, it is given that the LCM of x, 4^3 & 6^5 is 6^6.
we can write is like this way. LCM (x,2^6, 2^5*3^5) is 2^6*3^6.
Tha means that the LCM has 2^6*3^6 & we have 2^6, 2^5*3^5 therefore x must contain 3^6. alongwith 0,1,2,3,4,5,6. Therefore, x can be 7 different values. Therefore, 7. ........................C.
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 11:05
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Bunuel wrote:
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.

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
Kudos points for each correct solution!!!

---------------------------------------------------------------------------------------------------------------
Yeah, Here if we add 1/3 + 1/9 + 1/27 + 1/37 we will get the value as ::::: 0.50850850850850850850850850850851. it will keep on repeating itself in groups of three & @ 101th place it has a value. 0. Therefore, ......A.
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 11:17
Bunuel wrote:
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.

4. 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
Kudos points for each correct solution!!!

-----------------------------------------------------------------------------------------------------------------
It is given that F(n) = # of perfect Squares < n &
G(n) = # of primes less than n. & it is given that F(n)+G(n) = 16 so, the values in the two functions can be anything,
so to keep the options limited, lets look at the answer choices.
A. 30 < x < 36 .........
Perfect Squares < 30 = 1 4 9 16 25 i.e. 5.
Prime Numbers < 36 = 2 3 5 7 11 13 17 19 23 19 31 i.e. 11.
Therefore, this satisfies the condition given. Therefore,............. A.
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 11:21
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Bunuel wrote:
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.


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
Kudos points for each correct solution!!!

-----------------------------------------------------------------------------------------------------------------
The Q asks, How many of the subsets contains the five elements leaving behind 0.
so we can calculate the # of subsets as 2^5 = 32. Therefore, 32 . .........D.
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 11:30
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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.

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
Kudos points for each correct solution!!!

-------------------------------------------------------------------------------------------------------------
Here it is given that the GCF of two integers is 25 or 5^2. That means that they both share 5^2 only.
Now we can depict the intergers as 25a & 25b sharing only 25 & a & b don't share anything b/w them. okay !!
Now, as given in the question that sum of the integers is 350. therefore, 25a+25b=350 or 25(a+b)=350
or a+b =14. Now we look for those values of a & b those don't share anything b/w them. So,
a+b= 3+11 or 1+13 or 5+9.........so Three values. Therefore, 3. C.......................... :-D
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 14:59
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Question 1
Diagonals: 15x, 11x, 9x
Area of S: 225.x^2/2
Area of R: 99.x^2/2

Difference: 63.x^2

For I to be possible: x has to be equal to 1 (ok!)
For II to be possible: x has to be equal to sqrt(2) (NOT ok, since diagonals must be integers)
For III to be possible: x has to be equal to 2 (ok!)

Answer: D

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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 14:59
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Question 2:

Subset with 0 letters: 1
Subset with 1 letter: 7
Subset with 2 letters: C(7,2) = 21
Subset with 3 letters: C(7,3) = 7.6.5/6 = 35

Total: 1 + 7 + 21 + 35 = 64

Answer: E

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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 14:59
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Question 3:

Total number of subsets: 2^6 = 64
Number of subsets that include 0: 2^5 = 32

Total of subsets that do not contain 0: 64 - 32 = 32

Answer:D

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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 15:00
Question 4:
Easier if you test numbers (considering that f(x) + g(x) is always crescent).

x = 30 -> f(30) = 5, g(30) = 10 -> f+g = 15
x = 31 -> f(31) = 5, g(31) = 10 -> f+g = 15
x = 32 -> f(32) = 5, g(32) = 11 -> f+g = 16
x = 33 -> f(33) = 5, g(33) = 11 -> f+g = 16
x = 34 -> f(34) = 5, g(34) = 11 -> f+g = 16
x = 35 -> f(35) = 5, g(35) = 11 -> f+g = 16
x = 36 -> f(36) = 5, g(36) = 11 -> f+g = 16
x = 37 -> f(37) = 6, g(37) = 11 -> f+g = 17

Therefore: 30 < x < 37

Answer: B
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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 15:01
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Question 5:
Best solution is through elimination:
18! is divisible by 15, and 17. Therefore, 18! + 1 is not divisible by neither of these.
18! is also divisible by 33 as it has the factors 11 and 3. Therefore, 18! + 1 is not divisible by 33
18! is also divisible by 39 as it has the factors 13 and 3. Therefore, 18! + 1 is not divisible by 39

The only remaining option is: 19

Answer: C

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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 15:01
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Question 6
LCM (x,2^6,3^5.2^5) = (2^6).(3^6)

x = 2^a. 3^b
b must be equal to 6 (since 3^6 is a factor of the LCM)
a can be equal to 0,1,2,3,4,5,6

Therefore: 7 possibilities

Answer: C

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Re: Fresh Meat!!! [#permalink] New post 17 Apr 2013, 15:01
Question 7:

x + y = 350
x = 25*p, y = 25*q where p and q may not have any other factors in common

Substituting into the equation we have: p+q = 14
The only solutions for pair (p,q) in which p and q do not have common factors are: (1,13), (3,11),(5,9)

There are exactly 3 pairs: (25,325), (75, 275), (125, 225)

Answer C

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Re: Fresh Meat!!!   [#permalink] 17 Apr 2013, 15:01
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