-. Suppose that a positive integer is written . Prove that n is divisible by 9 if and only if y 9.

Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: Let A is multiples of 3 then show that it is countable infinite

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Q: Prove that the product of any k consecutive positive integers is an integer multiple of k!.

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Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

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Q: Find all integers z, positive or negatives, such that z2 + z is prime.

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Q: Find all integers 100 ≤ n ≤ 999 such that the last three digits of n² is same as n.

A: To find: All integers  such that the last three digits of is same as n.

Q: Compute for an arbitrary integer k

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Q: n - 2 + 2 k -1 n – 2) + k – 2) n n – 2 k k Give a double counting/combinatorial proof of the…

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Q: prove that for all integers m, n & d if d/m & d/n then d/(2m+3n)

A: We know that if a and b are integers such that a/b then b is a multiple of a. i.e. b=ka where k∈ Z{…

Q: 3. Let a and b be positive integers. Express the positive greatest common divisor of a and b in…

A: The positive greatest common divisor (gcd) of two positive integers, a and b, can be expressed in…

Q: 5. Show that 2 32 3" 31 3" for all positive integers n.

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Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: Prove that, all integers x and y, x^2-4y≠2  1a. Prove that all numbers of the form 1007, 10017,…

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Q: Prove that the sum of any five consecutive positive integers is always a multiple of five.

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Q: Prove that 3 divides n³ +2n whenever n is a positive integer.

A: Introduction: Consider a starting point for which the assertion is correct. The proposition is true…

Q: If the integer K is greater than or equal to 1 and is odd. Prove that the sum of the k powers of…

A: The N must be a perfect square or double square in order for the total of the k powers of positive…

Q: 2. Prove that the difference of an odd integer and an even integer is odd.

A: Prove that the difference of an odd integer and an even integer is odd

Q: Prove that 6">5" + 9 for any integer n ≥ 2.

A: Mathematical induction is a technique used to prove statements about well-ordered sets of natural…

Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

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Q: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

Q: ne order of the element 16+ in Z24/ is None of the choices 8.

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Q: prove that n²_1 is divisible by when ever n is odd an Positive integer.

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Q: Prove that if the sum of the digits is subtracted from a natural number, then the result is…

A: let us assume the number to be three digit number xyz therefore, xyz=100x+10y+z sum of digits will…

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: Prove that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

Q: For each pair of integers r and s below, Find the greatest common divisor d. Then, find m, n, such…

A: To determine the greatest common divisor, d and then to find m and n such that d = mr + ns. (a) r=32…

Q: 3. Show that there exist 100 consecutive positive integers that are not perfect squares.

A: Here we have to show that there exists 100 consecutive positive integer that are not perfect square.…

Q: 2. Use Fundamental Theorem of Arithmetic to prove that the product of any two odd integers is an odd…

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Q: Prove: Any positive integer can be represented as an aggregate of different powers of 3, the terms…

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Q: Prove that n2 + 4 is not divisible by 3 for all integers n.

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Q: Find all positive integers n such that the sum of all positive divisors of n is odd.

A: Given the data

Q: Prove that 74n+3 + 2 is a multiple of 5 for all non-negative integers n.

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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 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…