For every positive integer a, find a composite number n such that n|a" – a.

Q: An integer n is said to be abundant if o(n)>2n. Every multiple of abundant integer is abundant *…

A: We know,  abundant number is a number that is smaller than the sum of its proper divisors. An…

Q: Find an expression for An, where n is an arbitrary positive integer.

A: Given: A=1423∈M2×2R

Q: Suppose that the number 'a' has a property that for every natural number n, a ≤ 1/n. Prove that a ≤…

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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: Every integer is even or odd. Use induction.

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Q: Let x be the set of all positive integers and d(min) = | = - = | show that (x,d) is not complete

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Q: For each nonnegative integer i, what is the least residue modulo 9 or 10i?

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Q: show that for eury A s integer nz3; A n- Am-2 +A2J. Mence determine ASO.

A: To show: That An=An-2+A2-I for n≥3 then find A50. Given: The matrix is A=100101010. Calculation:…

Q: Let a and b be two integers such that b | a. Let p be a prime such that p|a and p{b. Prove that p|…

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Q: Prove that for any natural number N, there exists N consecutive integers none of which is a power of…

A: Please see the attachment

Q: Let a be a nonzero real number. Prove that aman = am+n for all integers m, n > 0.

A: Let a be a non zero real number , where m and n are integers…

Q: For every positive integer a, find a composite number n such that n|a^n - a.

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Q: •Prove: For all integers a, b, and c, if a|b and alc, then a|(b+ c). we can assume a #0.)

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Q: Prove: For all integers a and b, if a + b is odd then exactly one of the integers, a or b, is odd.

A: (1) We have to prove that a+b is odd then exactly one of the integer a or b is odd. Suppose that a…

Q: Find all integer numbers a for which

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Q: For each natural number n, let 1 n+1) = {r €R: <r< 2n+1} . An n2 Calculate (with proof) U An and N…

A: The objective is to calculate:

Q: Use mathematical induction to prove that the statement is true for every positive integer n.

A: Consider the given equation as abn=anbn

Q: For each positive integer n, find with proof GCD(220° - 220, 220" + 218).

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Q: Let a = 1, and let an+1 4a, – 1, for all nonnegative integers n. Prove that an 2-1"+1 3 %3D

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Q: For each positive integer n 2 0, let P(n) be the formula 3n+1 – 1 1+3 + 32 + ... 3n 2

A: We prove this formulas by using PMI

Q: Use mathematical induction to prove that n2- n is divisible by 2 whenever n is a positive integer. B…

A: Solve

Q: A(n) __________is a letter used in algebra to represent anynumber from a given set of numbers

A: In algebra, there are different set of numbers such as Real numbers, Complex numbers, Rational…

Q: Prove the statement for positive integer n and r, with r ≤ n. ( n n − r ) = ( n r )

A: We know that, the term nr for positive integer n and r, with r ≤ n is given as nr=Crnnr=n!n-r!r!…

Q: For an arbitrary integer, say A, verify that 2|A(A+1). Similarly, verify that 3|A(A+1)(A+2).

A: To verify:A(A+1) is divisible by 2.

Q: Find all n such that σ(n) = 72.

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Q: Find the smallest integer i for which the statement (1 + i2 )<2' is true. Then, prove the statement…

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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: For any positive integers m, n (

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Q: Show by induction that for each positive integer k, 1?+ 2² + 3² + . . · +k² = }k(k+ 1)(2k + 1).

A: NOTE: Refresh your page if you can't see any equations. . check for k=1 the statement is true…

Q: Find an explicit formula for an if a0= 1, a1= 2, and an= 2an−1−an−2for n ≥ 2.

A: given a0= 1, a1= 2  and an= 2an-1−an-2for n ≥ 2.

Q: Show that if n is a positive integer, then (n + 1, n2 - n + 1) = 1 or 3

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Q: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

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Q: For each positive integer n, show that и(п)и(n + 1)u(п + 2)u(n + 3) — 0

A: To prove : μ(n) μ(n+1) μ(n+2) μ(n+3) = 0  for each positive integer n.

Q: A sequence {am} of real numbers is defined recursively by a1 = 1, a2 = 2 and a, = (i–1)a; for n > 3.…

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Q: Let p = p1 ... Ps be an integer with digits p1,..., Ps. Show that 0.p1 . . · Ps %3D 10° – 1

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Q: If a is an integer such that: 2 ła, 5ła Show that (a)⁴ⁿ = 1(mod 40) for any positive integer n

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Q: For all positive integers a, b, where a < b, if a {b, then gcd(a, b) = 1. O True False

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Q: For every natural number n, one of n-1, n, or n+1 is a multiple of 3

A: The following two steps are used to prove a statement P(n) by the principle of mathematical…

Q: Let i be a positive integer and A, = [-i+1,1[. Then N14= %3D

A: We have to find the first intersection first.

Q: Let n be a positive integer and M %3D Then M"

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Q: Prove that for every n∈ N is a2/(a+1)n+1-na-(a+1), a∈ N

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Q: Let n > 1 be an integer. Show that if pn for all primes p≤ √n, then n is either prime or the product…

A: Introduction: Proof by contradiction, sometimes referred to as indirect proving or the reductio ad…

Q: Find all positive integers n such that p(n)|n.

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Q: Let n and r be non-negative integers such that r <n. Then, C + "C , = **C,

A: nCr=n!r!(n-r)!

Q: Let a, be the n th digit after the decimal point in 2n + 6e. Evaluate an(.1)". n=1

A: in given problem we have to evaluate :

Q: Show that if a and b are integers such that a|b, then a^k|b^k for every positive integer k.

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Q: m above, prove that for every real number c>0, there is a 1/N < c.

A: In every real number c, there is integer N which is N &gt; c.

Q: Find an explicit formula for an if a0= 1 and an= (n + 1)an−1+ 3^n for n ≥ 1

A: Answer