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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

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

4. Pick the numbers. Let's check the smallest possible: x=31. f(31)=5 (1,4,9,16,25) and g(31)=10 (2,3,5,7,11,13,17,19, 23,29). Therefore f(31)+g(31)=15, that is not equal to 16.

But if we take x=32: f(32)=5 and g(32)=11 (10 previous and 31). So, x=32 is a possible value. Next perfect square of an integer is 36 and the next prime number is 37. So, for not changing the number 16, we can take x=33, 34, 35, 36.

The correct answer is C. _________________

I'm happy, if I make math for you slightly clearer And yes, I like kudos:)

5. The number 18!+1 is not divisible by any number from 2 to 18, because 18! is divisible by them. So, the answers A and B are not correct. Since 18!+1 is not divisible by 3, 18!+1 is not divisible by 33 and 39. So, answers В and E are wrong

The correct answer is C. _________________

I'm happy, if I make math for you slightly clearer And yes, I like kudos:)

6. The problem states that lcm(x, 2^6, 2^5*3^5)=2^6*3^6. When we calculate lcm we have to take the highest powers in prime factorizations of numbers. If lcm contains 3^6 it must be in some number. So, x=3^6*y. y could be any factor of 2^6. The possible values of x: 3^6, 3^6*2, 3^6*2^2, 3^6*2^3, 3^6*2^4, 3^6*2^5, 3^6*2^6.

The correct answer is C. _________________

I'm happy, if I make math for you slightly clearer And yes, I like kudos:)

7. Let two numbers be 25n and 25m, where gcd(n,m)=1. Then 25n+25m=350 or n+m=14. So, our goal is to find pairs of numbers (n,m) such that gcd(n,m)=1 and n+m=14: (1,13), (3,11), (5, 9).

The correct answer is C. _________________

I'm happy, if I make math for you slightly clearer And yes, I like kudos:)

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

Let’s say diagonal of square is 15x and diagonals of rhombus are 11x and 9x where x is an integer. Area of square = (1/2) * 15^2 * x^2 = (1/2) * 225 * x^2 Area of rhombus = (1/2) * 11*9 * x^2 = (1/2) * 99 * x^2 Difference of areas = (1/2) * 126 * x^2 = 63 * x^2 So, the different must be a multiple of 63.

I. Here x^2 = 1. Possible option. II. 126 = 63*2. Here x^2 = 2. So, x is not integer. This option is not possible. III. 252 = 63*4. Here x^2 = 4 --> x = 2. Possible option

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

Number of subsets with 0 letter = 1 Number of subsets with 1 letter = 7C1 = 7 Number of subsets with 2 letters = 7C2 = 21 Number of subsets with 3 letters = 7C3 = 35 Total number of subsets with at most 3 letters = 1 + 7 + 21 + 35 = 64

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 we need to find out the possible subsets with the numbers {1,2,3,4,5}.

In the short way: Total number of subsets = 2^5 = 32

In the long way: Number of subsets with 0 element (null set) = 1 Number of subsets with 1 element = 5C1 = 5 Number of subsets with 2 elements = 5C2 = 10 Number of subsets with 3 elements = 5C3 = 10 Number of subsets with 4 elements = 5C4 = 5 Number of subsets with 5 elements = 5C5 = 1 Total number of subsets = 1+ 5 + 10 + 10 + 5 + 1 = 32

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

For x = 37, f(x) + g(x) = 5 + 12 = 17 For x = 36, f(x) + g(x) = 5 + 11 = 16 For x = 32, f(x) + g(x) = 5 + 11 = 16 For x = 31, f(x) + g(x) = 5 + 10 = 15 So, 31 < x < 37

18! and 18!+1 are consecutive integers and so they do not have any common factor except 1. 15, 17, 33 (=3*11), and 39 (=3*13) are factors of 18! and none of those can be a factor of 18!+1. So, only 19 can be a factor of 18!+1

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

350 = 25 * 14 To have the GCD of two numbers to be 25, we need to split 14 into two co-prime numbers. Such pairs of numbers are: (1,13), (3,11), and (5,9).

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

x^y=1 can be obtained by any of the following: (a) y = 0 and x = any real number (b) x = 1 and y = any real number (c) x = -1 and y = any even number

I. For x=2 and y=0, x^y=1. So, this condition is not necessarily true. II. For x=2 and y=0, x^y=1. For x=1 and y=5, x^y=1. So, this condition is not necessarily true. III. For x=-1 and y=2, x^y=1. So, this condition is not necessarily true.

Q4). 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: Answer - C

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 My answer is D Let the diagonals be 15x,11x and 9x. Area of a Rhombus is .5*d1*d2 .5*11x*9x=> 11*9*x2/2. = (99x^2)/2 Area of square with diagonal 15x is (225x^2)/2 Difference in areas is 63x^2. Difference can be 63, for X = 1 ; 126 for x=root2 and 252 for x=2. However when x= root2, the diagonals are not integers. Hence only 2 values are possible. 63 and 252

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 My answer C Squares less than 36 : 1,4,9,16,25 = 5 primes less than 36 : 2,3,5,7,11,13,17,19,23,29,31. =11 and sum =16. Hence the max value N has to be less than 37 as N =38 will increase the number of primes by 1 and squares by 1. And the sum will be 18 Minimum values of N has to be 32, as any value less than 32 will decrease the number of primes by 1.and the sum will be 15

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