. Let x be a positive real number and d be a positive integer. Prove that the number of positive integers less than or equal to x that are divisible by d is [x/d]. 2. Find the number of positive integers not exceeding 500 that are divisible by 3.  3. Find the number of positive integers between 200 and 500 that are divisible by 3.

Q: 74. Prove that for any real numbers a and b, |b − a|=|a − b|.

A: Given; a and b are any real numbers Consider, L.H.S=b-aWe can write it as,=b+-a=-1-b+a=-1a-b…

Q: 2. Let x1x₂ >. (1+x₁) n . xn> 0 ben real num 1 + x₁ + x1x2 + .. + x1xz

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Q: Ex: Let X₂ [1,2,3,4,5]. I. [0,X, [1], 82.3, 81.23], Y. [3,4,5,6,73 andl T₁: [0, X, [4], [4,633 if A:…

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Q: For every positive 2-digit number, x, with tens digit / and units digit u, let y be the 2-digit…

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Q: = Let a, b, and d be positive integers. Prove that if (a, b) : divides a, then (d, b) = 1.

A: Definition of gcd (a, b) =1 is a and b does not have common prime divisor. Using this we can easily…

Q: Example 5-42. (a) Find the Geometric Mean of 2, 4, 8. 12, 16 and 24. (6) If the observations 2, 4, 8…

A: HEre AS PER POLICY I HAVE CALCULATED FIRST MAIN QUESTION  geometric mean =  a1 *a2*......ann

Q: 2. The set of all rational numbers in the interval (0, 1).

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Q: Give the greatest integer function of [5.6]

A: The greatest integer function : The greatest integer function is a function that rounds up the…

Q: 4. Let h(r) = 5 x 6 20 x²3x18 Find all values of N for which h(N) = 3.

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Q: FIND AUB,GIVEN THAT A= {x/x is a positive integer. 2<x<6}    B= {x/x is a negative integer, - 6<x<…

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Q: 1. Prove that the sum of an integer and its square is even

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Q: 3.54. If the arithmetic mean of two unequal positive real number 'a' and 'b', (a > b), be twice as…

A: In question, Given that the arithmetic mean of two unequal real numbers a and b be twice of their…

Q: 10. Let x be a real number. Prove that at least one of the numbers /2 – x or /2 +x is irrational.

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Q: Solve 2|7-x| 8 c. -8 6

A: We have given the inequality; 2|7-x| &lt; 2

Q: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

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Q: 2 4. Express 989 = 23 43 in the form x² + xy + 5y².

A: The given number is 989. The objective is to express the given number in the form of x2+xy+5y2.

Q: 7. Let y be irrational and x ‡ 0 a rational number. Show that the product xy is irrational.

A: We will prove given statement by method of contradiction.

Q: Determine the union X ∪ Y. Express them in interval notation. X = {x/x>-17 } and Y = {x/x<equal-10…

A: The union of two sets A and B is defined as the set of all the elements which lie in set A and set B…

Q: 3. Find all r so that r-1+r-2| < 5.

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Q: Determine whether the following statements below is true or false. 1. (1/2)2<1/2, and 123 is…

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Q: Find the total number of integers 'n' such that 2 ≤ n ≤ 2000 and H.C.F. of n and 36 is 1.

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Q: a. 2 |x – 4|| х-1 b. x? + 2х — 12 1 х+1

A: a.  (1, 2] and [3, (5+√17)/2]         b.  (-1-√13 , -1+√13)

Q: 2.11. Show your steps. a) Find all integers x such that 4x =11 9. b) Find all integers z such that…

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Q: Determine the union X ∪ Y. Express them in interval notation. X = {x/x>-17 } and Y = {x/x<equal-10…

A: Introduction: The number of elements carried by the operation and executing the results of a…

Q: Q2 1-The Euler's number (e) is a * rational number a. True b. False

A: Rational numbers are those which can be represented in the from of fraction and those which can not…

Q: Let Find formulas for the entries of M", where n is a positive integer. Hint: a formula such as 5…

A: The given problem is to find the entries of Mn for the given following matrix M. We have to find the…

Q: Prove by contradiction method : if x>0  then 2 ≤ x + 1/2

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Q: Let Z, = {0,1,2,3,4, 5,6} where each x € Z, is a least residue when an integer is divided by 7. %3D…

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Q: 2. Determine all values of x - y when x and y are integers such that x = 360 + y².

A: The given equation is. x2=360+y2x2-y2=360x+yx-y=360 Since it is given that x and y are integers, so…