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# Fresh Meat!!!

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Fresh Meat!!! [#permalink]  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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Re: Fresh Meat!!! [#permalink]  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

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Re: Fresh Meat!!! [#permalink]  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)

Current Student
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Re: Fresh Meat!!! [#permalink]  17 Apr 2013, 15:02
Question 8
x.377910 = 3300k

=> x.(19.17.13.5.3^2.2) = 11.5^2.3.2^2.k
=> x. 19.17.13.3 = 11.5.2
=> x = 110.k/(19.17.13.3)

Therefore, the smallest number for x is 110 (when k = 19.17.13.3)

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Re: Fresh Meat!!! [#permalink]  17 Apr 2013, 15:03
Question 10

1. x = 1 ? Not necessary. Let x = 3, and y = 0, then x^y = 1
2. x = 1 and y = 0. Not necessary (see item 1)
3. x = 1 or y = 0. Not necessary. Let x = -1, y=2

Last edited by caioguima on 18 Apr 2013, 03:48, edited 2 times in total.
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Kudos [?]: 51 [0], given: 27

Re: Fresh Meat!!! [#permalink]  17 Apr 2013, 17:34
Question 5

I get that the answer is 19 that is C, but i am unable to find out how!
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Re: Fresh Meat!!! [#permalink]  18 Apr 2013, 00:34
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

Sol: We need to find condition which will be true under all conditions

St 1 when x =1 then X^ (any value of y) will always give us 1. So A, D are possible answers
St 2 X=1 and y=0, Here and means both conditions simultaneously. But Y need not be zero as long as x= 1. Y can take any value as long as x=1 and hence B is ruled out

St 3 x=1 or y=0 here "or" means any one of the condition if true then we get x^y=1 which is true. Consider x=1 and y= 32 ----> x^y = 1, consider x= 1, y= -3, then x^y= 1
Similarly X= 2 but y=0 then 2^0 =1 Option C and D can be the possible answer

Ans Option D......

PS: I think there is a catch since St 1 covers only value of x whereas St 3 covers for all possible cases of (x,y) and I was tempted to go for option C alone as the answer. Let's see
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Posts: 729
Location: India
GPA: 3.21
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Kudos [?]: 534 [0], given: 723

Re: Fresh Meat!!! [#permalink]  18 Apr 2013, 01:06
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

Sol: 1/3 + 1/9 + 1/27 + 1/37

1/3= 0.333333
1/9= 1/(3*3)= 0.3333333/3-----> 0.1111111
1/27= 1(3^3)= 0.037037037
1/37= 0.027027027
0.333
0.111
0.037
0.027
Sum is ( .508508508)...99th Digit will be 8,100th digit will be 5 and 101st digit will be 0...

I guessed it under timed condition......Would like to have faster way.

Ans should A
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Joined: 07 Sep 2010
Posts: 335
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Kudos [?]: 401 [0], given: 136

Re: Fresh Meat!!! [#permalink]  18 Apr 2013, 21:49
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

All are Could be true questions, but none of them is MUST be TRUE\
1- X can be any other value than 1
2- NoT Necessarily
3- x=2 and y =0; x=1 and and y=1 can satisfy
Hence, E
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Re: Fresh Meat!!! [#permalink]  18 Apr 2013, 21:52
5. Which of the following is a factor of 18!+1?

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

Since 18! must result in 00
And adding 1 into it = the last two digits become 01

Only multiple of 17 has 01 as last two digits.

Hence, B
Not too sure though
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Last edited by imhimanshu on 18 Apr 2013, 22:39, edited 1 time in total.
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Re: Fresh Meat!!! [#permalink]  18 Apr 2013, 21:54
Q5) Which of the following is a factor of 18!+1?

18! + 1 is only 1 more than the 18!, so none of the factors - 2, 3, 4, 5, ...,15, 16, 17, 18 of 18! would be factors for 18! + 1
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Kudos [?]: 52 [0], given: 65

Re: Fresh Meat!!! [#permalink]  18 Apr 2013, 22:02
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

1^y is always 1 and x is not equal to 0

x^0 is not always 1 especially when x is 0 - undefined
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Last edited by nt2010 on 19 Apr 2013, 13:32, edited 1 time in total.
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Re: Fresh Meat!!! [#permalink]  19 Apr 2013, 01:53
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

Here, should we include empty set?
Math Expert
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Kudos [?]: 62511 [0], given: 9457

Re: Fresh Meat!!! [#permalink]  19 Apr 2013, 04:48
Expert's post
rbansal6 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

Here, should we include empty set?

Yes, empty set should be included.
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Re: Fresh Meat!!! [#permalink]  19 Apr 2013, 13:29
Q7) 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?

Since the GCM is 25, the 2 positive integers should have 5^2 as the only common factor. The 2 potential positive integers can be found by dividing 350 / 25 and finding unique prime factors that add up to the divisor.

350 / 25 = 14 and 14 = 11 + 3
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Re: Fresh Meat!!! [#permalink]  19 Apr 2013, 14:04
Q1) 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

S diagonal : R diagonal 1 : R diagonal 2 = 15:11:9

Let's assume x is the unknown multiplier and the ratio of diagonals are 15x:11x:9x

$$Area of S =\frac{15x}{\sqrt{2}}$$

$$Area of R = 1/2 * 11x * 9x$$

$$Area of S - Area of R = \frac{225x - 99x}{2} = 63x$$

And I, II, II are multiples of 63 when x = 1, 2, 3
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Re: Fresh Meat!!! [#permalink]  19 Apr 2013, 23:08
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
Let the numbers be 25a and 25b. for these numbers to have 25 as the GCD, a and b must be co-prime i.e have only 1 as the common factor.
25a+25b=350 => a+b =14 , a and b are co prime. Hence the pairs of a and b are => (1,13) , (3,11) ,(5,9).
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Re: Fresh Meat!!! [#permalink]  19 Apr 2013, 23:08
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

(337910*x)/3300 , reduced will become (12667 *x)/110 and 12667 is not divisible by any of the factors of 110( 2 or 5 or any combinations of both). Hence X has to be 110. At least.
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Re: Fresh Meat!!! [#permalink]  19 Apr 2013, 23:09
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

1/3 + 1/9 + 1/27 = 13/27. = .481481…
1/37=.027027…
13/27 + 1/27 = .508508..
Hence 101st digit is 0.(3*33 +2=> hence the second digit of the recurring decimal which is 0)
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Re: Fresh Meat!!! [#permalink]  19 Apr 2013, 23:17
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
Since the question is what MUST be true I would go with C. as X can be any integer apart form 0 and Y can be 0. It’s not a must for X=1 and y=0.
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Re: Fresh Meat!!! [#permalink]  21 Apr 2013, 22:27
Expert's post
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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

Given: $$\frac{377,910 *x}{3,300}=integer$$.

Factorize the divisor: $$3,300=2^2*3*5^2*11$$.

Check 377,910 for divisibility by 2^2: 377,910 IS divisible by 2 and NOT divisible by 2^2=4 (since its last two digits, 10, is not divisible by 4). Thus x must have 2 as its factor (377,910 is divisible only by 2 so in order 377,910*x to be divisible by 2^2, x must have 2 as its factor);

Check 377,910 for divisibility by 3: 3+7+7+9+1+0=27, thus 377,910 IS divisible by 3.

Check 377,910 for divisibility by 5^2: 377,910 IS divisible by 5 and NOT divisible by 25 (in order a number to be divisible by 25 its last two digits must be 00, 25, 50, or 75, so 377,910 is NOT divisible by 25). Thus x must have 5 as its factor.

Check 377,910 for divisibility by 11: (7+9+0)-(3+7+1)=5, so 377,910 is NOT divisible by 11, thus x must have 11 as its factor.

Therefore the least value of x is $$2*5*11=110$$.

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Re: Fresh Meat!!!   [#permalink] 21 Apr 2013, 22:27

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