please help me with these questions mere dosto?

Question

3. What is the two-digit number whose cube root is the square root of the sum of its digits?

5. What is a number that is equal to the sum of the digits of its cube?

11. What is the only prime that is the sum and difference of two primes?

13. What is the radius of a circle when the numeric values of the circumrence and the area are equal?

16. What is the only perfect number of the form x^n+y^n?

18. What is the smallest number of people in a room where the probability of two of them having the same birthday is atleast 50%?

20. What is the only integer n for which n = x^n = y^n, for x ≠ y?

26. (I know I already asked this one but I need a whole number as an answer) What is the 1/2+1/4+1/8...

32. What is the smaller of two triangular numbers whose sum and difference are also triangular numbers?

54. What is the only cube that is less than a square?

55. What is one of the two numbers that can be both the perimeter and the area of the same rectangle?

Answer 1

3. 27
11. 5
13. 2
20. 0
26. 1 (actually, it's a limit that approaches 1)
54. The cube of any negative number.
55. 4 * 4 = 16

Answer 2

3 No. is 2 7 cube root = 3 is sq.rt. of 2+7 i.e. 9
5 No. is 8 cube = 512 8 = 5+1+2
11 No. is 5 = 3 +2 and = 7 – 2
13 circumference = area of circle
2 pi × r = pi r^2 therefore radius r = 2
16 If the sum of divisors of number, excluding the number is equal to the number is perfect number.
Number is 28 = 3^3 + 1^3
Divisor are 1, 2, 4, 7, 14 & 28
Sum of divisors except 28 is 1+2+4+7+14 = 28

18 The smallest number is Only 2 out of these 50% i.e. one will have same birthday.


20

26 1/2 = 1 – 1/2, 1/4 = 1/2 + 1/2 1/8 = 1/4 + 1/4 and so on
when you add this all the term gets cancelled except 1
Therefore ans is 1
32 Triangular numbers are represented as n(n+1)/2
when n = 3, Triangular No. = 6, When n = 5 triangular No. is 15 and when n = 6 Triangular No. = 21 when n = 7 Triangular No. = 36
15 and 21 difference 21 – 15 = 6 and 15 + 21 = 36
Out of 15 and 21 , 15 is the smaller.

54 cube of any – ve number is less than square of that number

55 The number is 18 where L = 6 and B = 3
Area = 6×3 and perimeter = 2(6+3) =18
A square is also a rectangle therefore square with side 4 is having perimeter and area same.

Answer 3

3. 27 will work. 27^(1/3) = 3 = 9^(1/2) = (2+7)^(1/2).

5. 8. 512 = 5+1+2=8.

11. 5 is the only such prime: 5 = 3+2 = 7-2.

13. 2πr = πr², so r²=2r. The only positive
solution is r = 2.

16. Well, 28 = 3³ + 1³. I don't know if this is the
only such solution.

18. The answer is 23. See the reference below
http://en.wikipedia.org/wiki/Birthday_paradox

20. 0. x^0 = y ^0 = 1 for all x, y not 0.

26. This is an infinite geometric series which
converges to 1. The formula for the sum is 1/2 /(1-1/2) = 1.

32. 15. The larger number is 21.
21+ 15 = 36 and 21-15 = 6 are both triangular numbers.

54. This question has infinitely many answers.
Any positive integer cube other than 1 is less than its square.
I.e., The sixth power of n(n>1) is bigger than n³.

55. p = 2l+2w = lw
Certainly 16 will do if l and w are both 4.

Answer 4

3. 27
5. 1
11. 5 (2+3) (7-2)
13. radius of 2, circumference would be 4pi area 4pi^2
16. 25 it is 5^2 and the sum of 3^2 and 4^2
18.
20.
26. 1
32.
54. the cube of any fraction will be less than its square
55. 16 (a square with a side length of 4)

Answer 5

3. 27.....27^1/3 = (2 + 7)^1/2
5. 8.....8^3 = 512, 5 + 1 + 2 = 8
11. 5.....2 + 3 = 5, 7 - 5 = 2
13. 2.....PiR^2 = 2PiR, cancel PiR from both sides, R = 2
16. ???...I'm going to defer to Jeffrey for this one.
18. 23 see: http://en.wikipedia.org/wiki/Birthday_paradox
20. 0...everything to the zeroth power equals 1
26. approaches 1
32. 15...21 - 15 = 6, 21 + 15 = 36
54. Any number less than 1 will satisfy this (1/2, -3, etc.)
55. 16 and 18...sides 4x4 and 6x3

Answer 6

3) ???

5) ???

11) 5 (Sum primes is 3 and 2) (Difference between 7 and 2)

13) 2 (as pi r^2 = 2 pi r in this case)

16) ???

18) I think it is 183

20) n = 0 (as anything to the power zero = 1)

26) The series will end up being 1.

32) ???

54) Answer-: any number less than 1

55) ???

Answer 7

3. 27
5. 1
11. 5
13. 2
16.
18.25
20. 2
26. 1
32.
54. the cube of any fraction or negative no.will be less than its square
55. 16

Answer 8

3) 26
5) 8
11) 5 = 2+3 = 7-2
13) 2
16) 28 = 1^3 + 3^3
18) 23
20) 0
26) 1
54) (1/2)^3<(1/2)^2, ther are many...
55) 3,6

Answer 9

beats me!!!!!!!(not kidding..).