86- The remainder upon the division of x 98 + 4 by x – 4 in Z, is:

Q: Prove that 6 divides n^3 − n whenever n is a non-negative integer

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Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: (b) Prove that (a, b) = (b,r).

A:

Q: 2. Find six rational numbers between 3 and 4

A: We can solve the given problem using concept of rational numbers which is, Therefore d=(y-x)/(n+1)…

Q: 1. Let a, b E Z such that the remainder in the division of a by b is 140. If 3b has remainder 105 in…

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Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

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Q: Prove that 5 does not divide 5n+3 for any integer n.

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Q: Let a,n∈N. Assume that n≥2. Show that if a^(1/n) is rational, then it is an integer.

A: The detailed solution is as follows below:

Q: 2. Prove that the number of irreducible fractions not greater than 1 and with denominator equal to n…

A: For the solution of the problem follow the next step.

Q: 1. Which operation below does not change the value of any nonze ro number on oob A. Adding by One B.…

A:

Q: (a) In (x - 3) = 7 (b) In (x + 4) + In (x- 4) = 1

A: Given(a) In(x-3) = 7(b) In(x+4) +In(x-4) =1solve for x(a) In(x-3) = 7In(x-3) =7take Ine  both…

Q: Show that if any eight positive integers are chosen, two of them will have the same remainder when…

A: Pigeonhole Principle- If n items are assigned to m containers, and m<n,then at least one…

Q: 6. Find the remainder of 17036543 on division by 25.

A: Thus, remainder = 2

Q: Show that n22-1 is division by 8, if n is an odd positive integer

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Q: Show that n2-1 is division by 8, if n is an odd positive integer

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Q: 1- Show that if a is an integer, then 3 divides a^3-a. with details plz

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Q: When x is divided by 7, the remainder is 5. What is the remainder when 5x is divided by 7? Moreover,…

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Q: 32. Prove that 3 divides n + 2n whenever n is a positive integer.

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Q: 5) Using Fermat's little theorem, determine 21998 mod 7. In other words, what is the remainder when…

A:

Q: Show that there is no positive integer n for which In-1+ In+1 is rational.

A:

Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: 1. What is the remainder when 65544 - 13 is divided by 8?

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Q: Suppose multiplication over rationals was defined as follows:

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Q: 5. Given a real number x and a positive integer k, how many multiplications are used to find kx by…

A: Given,A real number x and positive integer kTo find: Total number of multiplications used with the…

Q: Prove that 6-1 is divisible by 5 for any n 21.

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Q: Show that zero is the identity for addition on R and 1 is the identity formultiplication on R. But…

A: addition:

Q: 29 Using the combination formula expand (5+x)

A:

Q: Find the smallest positive interger x such that 1001 x leaves a remainder of 70 when divided by 315

A: The given statement can be expressed as an expression, which ix given below  315y+70=1001x Now,…

Q: If n = 4k + 3, does 8 divide n2 - 1? (Assume that n and k represent integers.) O Yes, because n2 - 1…

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Q: Prove that 4 does not divide n2 +2 for any integer n.

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Q: Prove that n² +n is divisible by 2, where n E N.

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Q: Let a be an integer such that the remainder of dividing a by 17 is 15. What is the remainder of…

A: We will find out the required value.

Q: 3. Use Euler's Theorem (and the Chinese Remainder Theorem) to how that n12 = 1 (mod 72) for all (n,…

A: The Euler's Theorem states that for a positive integer n and an integer a that is relatively prime…

Q: Prove that n2+n  is  divisible by 2, where nN.

A:

Q: Prove that 11 divides 3* 5k + 42k+1 + 33k+2 – 5 * 72k for every k non-negative integer

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Prove: If the sum or the difference of two irreducible fractions is an integer, the denominators of…

A: Proof: Let two irreducible fractions are ab and cd Since the fractions are irreducible, (a,b)=1 and…

Q: "For all integers n, n(n² – 1) is divisible by 6." (You may use quotient-remainder theorem and…

A: Let us consider, for all integers n a = n3 – n a = n (n2 – 1) a = n (n + 1)(n – 1)

Q: 1. Suppose that you take an integer, n, apply the Quotient-Remainder Theorem where d = 7, and find…

A: The quotient-remainder theorem is expressed as n=dq +r and 0≤r<d where qand r are quotient and…

Q: Prove that 21 divides 2n-) 4nt whenever is Posi tive in teger in

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Q: Prove that 18 divides 2. 192n+1 16 . 174n+1 for all n e N.

A: We need to show that 18 divides 2·192n+1-16·174n+1 Now, a divides b is there exists an integer t…

Q: Prove 8 divides 34h+1+52h+1 whenever h is a positive integer

A: We need to prove that 8 divides 34h+1+52h+1 whenever h is a positive integer by Mathematical…

Q: Prove that 3" + 1 is divisible by 4 whenever n is an odd positive integer.

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Q: 8. Show that no prime number of the form 4k +3 can divide a number of the form n2 + 1.

A:

Q: Show that n22-1 is division by 8, if n is an odd positive integer

A: If n is a odd positive integer n can be written as   n = 2k+1 by substituting in n2-1 n2-1 =…

Q: Obtain the expansion of : 2 up to the term in x8 1-3x2

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Q: 5 Let flx) -5X+3Xh emd gx) = 3X+2X+) in 7[x].Determine the quotient ane remainder upon dividing #00…

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Q: Prove that for a positive integer n we have that Bn+3 - 26n – 27 is divisible by 169.

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Q: Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4. In…

A: Introduction: Divisibility is a property of an integer. For two integers a and b with b>a, the…

Q: Show that 19 divides 10*8n-1 + 33n-1 for every positive integer n.

A: Concept: Mathematical induction: For each and every natural number n, mathematical induction is a…

Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: