. 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: Prove thq+ for all integer a, b, and c if 9+b of a and k is even.

A: Introduction: Even and odd numbers are also referred to as: Even Number: An even number is one that…

Q: se for all integers nz2, tity = to shaw that for all integer

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Q: Find the least integer n such that (2x^3+x^2logx) is O(x^n).

A: The given expression is 2x3+x2logx. Concept used: Let f and g from the set of integers or the set of…

Q: 3. Let n and m be positive integers. Evaluate > (km+1 (k – 1)m+1) if possible. If not possible, k=1…

A: We have to evaluate the given summation.

Q: (e) For any nonzero real number z, prove Σ''-1 1-z"

A: The objective is to prove that for any nonzero real number x, ∑i=1nx-1xi=1-x-n

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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Q: Suppose there is a real number r with the property that r + 1/r is anodd integer. Prove that r is…

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Q: Let w be defined as 2 more than the number of digits in the integer w. For example, 15 =4(2 digits…

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Q: 5. Let a be a real number, c an integer, c > 0. Prove that

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Q: 4. Prove that if m and n are positive integers and x is a real number, then %3D m

A: We know ⨽x⨼ = x if x is an integer. Here, since m and n are integers, we have the following steps:…

Q: Let filN >N be defined by f(n) = Ź v(d) for all nEl din where r(R) is the number of the positive…

A: A multiplicative function on the set of natural numbers is a function T such that Tnm=T(n)T(m)…

Q: Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

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Q: 42 Show that E a Imod n) for all integer aif and only if n is squarefree. Ep(n) is Euler Totient…

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Q: Let x,y,z∈ Z be integers. Prove that if x(y+z) is odd, then x is odd and at least one of y or z is…

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Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

Q: Let a and b be two integers. Prove that if a+b is even then a-b is even.

A: Given that a and b are any two integers. We are trying to prove "if a+b is even then a-b is even."

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

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Q: Suppose if p and q are odd integers with p!= q (p is not equal to q). Show that there is a unique…

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Q: Let A be the set of all positive integers whose last two digits form a number that is evenly…

A: We have the given condition as   Let us consider A be the set of all positive integers whose last…

Q: Let a and b be positive integers such that ax + by = 7 for some integers x and y. Then %3D

A: We have to choose  current answer of given statement.

Q: Suppose that J and K are points on the number line. If JK= 10 and K lies at 3, where could J be…

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Q: integer of the form okts is also but not' conversely

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Q: Let a, b, c, d, e, and f be positive integers. Let d = gcd(a, b, c), e = gcd(a, b) and f = gcd(e,…

A: Given that for positive integers a,b,c,d,e,f , d=gcda,b,ce=gcda,bf=gcde,c

Q: Show that if x is an integer, then gcd (x+1, x^2 -x +1) divides 3. Use this result to find knowing…

A: Solving

Q: is the completion of CX sd) is the set of all rational number

A: We know that a metric space (X,d) is complete iff every Cauchy sequence in X is convergent in X but…

Q: . Use direct proof to prove that, if x is a positive odd integer, then z – 1 is wholly dividable у…

A: x^2-1 is divisible by 4.

Q: Let m and n be positive integers and let k be the least common multipleof m and n. Show that mZ∩ nZ…

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Q: Prove that if ni is a positive integer, a± 0

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Q: 3. Prove that if m is a positive integer and x is a real number, then 1 2 m 1 [mx] = [x] + \x + = |…

A: The greatest integer function is one of the most important real valued number theoretic function.…

Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

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Q: 2. Show that if a is an integer other than 0, then a) 1 divides a h) a divides 0

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Q: 4. Let R be the set of all real numbers, xER and n E N such that n 2 2. If (1+x) > 0, prove that P,…

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Q: 4. Demonstrate that for positive integers t and Q, (t <= [V@]) is equivalent to (t x t <= Q). Show…

A: Given that t and Q are two positive integers and t≤Q To show that t×t≤Q Since from the given…

Q: Prove that for every positive integer n the number (1 – /2)" is not ratioanal.

A: Given: The given number is, (1-2)n where n is the positive integer.   To prove: The number (1-2)n is…

Q: (b) Prove that 2m ≤ m! + 2 for m ¤ N. (c) For any nonzero real number x, prove ---3 i=1 = 1- x -n

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Q: Prove directly from the definitions that for all integers the Same parity Divide Bnt 19. and E m and…

A: If m an n are both even  let m= 2a and n= 2b   then  5m+7n=5(2a)+7(2b)=10a+14b=25a+7b therefore,…

Q: Prove that if x is real number then L2xd =LxJe bx+!]

A: We will prove the expression by two  steps taking real number x in two form.

Q: Suppose that c is a positive integer. Define f(c) to be the number of pairs (a, b) of positive…

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Q: coth- x = In () Q1/ Prove that

A: To prove coth-1x=(1/2)ln[(x+1)/(x-1)]

Q: b. H = {x E G|x" = e} for a fixed positive integer n.

A: Given set in part B

Q: Prove by contradiction that if a sum of three real numbers is less than 30, th at least one of the…

A: We have to prove by contradiction that if a sum of three real number is less than 30, then at least…

Q: Suppose that r, s are integers (possibly different from x, y res that ar + bs = c. Show that there…

A: Introduction: Let us suppose that two integers are a and b, greatest common divisor of these two…

Q: Prove that if æ is a positive real number, then LV=]] = LVE] %3D

A: Solution: Definition: The Floor function f is a function that takes any real number x and gives…

Q: Let x and y be positive integers. When is x + y3 a prime number? When is x+y a composite number?…

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Q: Show that 1 (N, <) is well-ordered. 1 2 (Z, <) is not well-ordered. 3 (R, <) is not well-ordered.

A: Please send questions as different questions. We cannot do more than 3 at a time.   A relation ≤ on…