Prove that if c is a natural number then for all a and b we have : gcd(ac, bc) = c gcd(a, b)

Q: Prove that for each natural number n, 5 divides n^4 − 1 if and only if 5 does not divide n. please…

A: Consider the given information. We have to prove that for each natural number n, 5 divides n^4 − 1…

Q: Suppose gcd(a, n) = 1. Show that there exists b such that [a]n * [b]n = [1]n.

A: Using division algorithm.

Q: 15. Prove that for each natural number n, 5+ ...+ (4n-3)(4n+1) 4n+1'

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Q: (5n)! 40"-n! is natural number for every natural number n. Prove that

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Q: Prove that for all natural number n, 52n-1 is divisible by 8.

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Q: Show that, for given integers a, b, c, if gcd(a, c) = gcd (b, c) = 1, then gcd (ab, c) = 1

A: Given a,b,c∈Z+ And gcda,c=1 and gcdb,c=1 We have to show that gcdab,c=1 Since, gcda,c=1 and gcdb,c=1…

Q: 3. Prove that for each natural number n, 十2 ...

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Q: Prove that for a positive integer n we have that  33n+3 - 26n - 27 is divisible by 169.

A: To show  33n+3 - 26n - 27 is divisible by 169.

Q: If z is an odd ratural number then 2^+1 is divisi ble by 3 hìnt: prove by Induction

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Q: Prove that for any integer n, n6k - 1 is divisible by 7 if gcd(n, 7) = 1 and k is a positive…

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Q: Prove that for a # -1, a>0,n EN IS (1+a)^ n >=1+na.

A: Here let A=(1+a)n, a≠-1, a>0, n∈N and B=1+na

Q: Let a, b, e, m, and n be integers such that aJb and a l e. Prove that a I (mb + ne).

A: Let a, b, e, m, and n be integers such that ab and ae. The objective is to prove that amb+ne. We…

Q: If a, b e Z such that a b then the numbers a and b are called relatively prime numbers if gcd(a, b)…

A: Consider the given statement. We'll prove the statement by contradiction. Let's the numbers are not…

Q: |Let a, b, and c be integers. Prove that if gcd(a, b) | bc, then a | 1 and gcd(a, c) 1 and a (a) C.…

A: To prove the required divisibility properties under the given conditions on the greatest common…

Q: 6. Prove by CVI that every natural number n > 2 is a product of prime numbers. NOTE. A prime number…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Prove that if for some integers a, b, c we have that a + b3 + c* is divisible by 9, then at least…

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Q: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

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Q: Let i be a positive integer and A, = [-i+1,1]. Then U2, A

A: We have to use properties of union in this question.

Q: Prove by Induction that if                                          a1,a2,a3,.........an belongs to…

A: I have proved for n=1, 2 and by assuming n=k i have proved for n=k+1

Q: 3. Prove that 2" > n For all nuatural numbers

A: Solution

Q: 4. Prove: if an/bn (Here, as always, L represents a finite number, not oo.) L, ba #0 for any n, and…

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Q: Prove that there is a natural number N such that for all natural numbers n > N, 2/n < 0.1.

A: Assume that there does not exist N such that for any natural number n,  n&gt;N, 2/n&lt;0.1.

Q: (a) Prove that for all n 2 0, Eo fi = fn+2-1 where fi-1 is the th Fibonacci number.

A: Fibonacci sequence

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

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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 the operation a*b=1+ab on the set of integers Z, has no identity element

A: Let e be the identity element which exist.  Then, a*e = e*a = a

Q: Prove by Mathematical Induction that for any natural number n the number 52n+1 – 22n+1 is divisible…

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Q: (1 + x)" > 1 + nx

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Q: 15. Prove that for each natural number n, s+ (4n-3)(4n+1) 4n+1 ...

A: Given The given expression is 11·5+15·9+·  ·  ·+14n-34n+1=n4n+1

Q: Show that the axiom A12' implies axiom A12, that is, if A12' is true and {A, B} be a Dedikind's cut…

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Q: Prove there are no natural numbers n and p with n > 2 and p² + 1 = 2".

A: ANSWER:

Q: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

Q: If nis a positive integer 1 then prove (D - a) n!

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Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: Prove that if a > 0, k > 1, and ak -1 is prime, then a = 2 and k is prime. %3D

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Q: Prove that every natural number n can be written in exactly one of the following forms: • 3k for…

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Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

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Q: . Prove that for each natural number n, /3n +2 is not a natural number.

A: Let n be any natural number.  n=3m+k2   =3m2+6m·k+k2   =9m2+6mk+k2   =33m2+2mk+k2 Here, m and k are…

Q: Prove by CVI that every natural number n > 2 is a product of prime numbers. NOTE. A prime number p…

A: Every integer greater than 1 can be written as the product of prime numbers. In other words, every…

Q: Prove by Induction that for every natural number п, 3h-1 is a multiple of 2?

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Q: Suppose B is a set. Use mathematical induction to prove that for every positive integer n and any…

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Q: (5n)! 40 n! is natural number for every natural number n. Prove that

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Q: (a) Determine the set of values of n for which and are positive integers. A 'Pythagorean triple' is…

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Q: 16. Prove that, for integers a and b, gcd(a, a + b) gcd(a, b). [Hint: Mimic the proof of Lemma…

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Q: Every natural number n has a base-60 expansion; that is, there exists k > 0, and for each i = 0, 1,…

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Q: Prove that there is an infinity of natural numbers n such that 2n +1 is divisible by n3

A: Solution of the problem as follows

Q: Prove that each statement is true for all natural numbers. 72 2n(n + 1)(2n +1) (2i)° : %3D 3. k=o

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Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…