Exercise 6.4 (1) For which natural numbers n is the number 3" + 1 divisible by 10? Find the remainder of the division of 1!+2!++50! by 7. (3) Is it true that 36 divides n¹ + n² +4 for infinitely many natural numbers n? Explain!

Q: 1 For any integer k, 3|(2k² + 1) if and only if 3 k

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Q: Prove the following statement or disprove by giving a counterexample "for all positive intergers n…

A: We have to prove or disprove the following statement For all positive integer n and for all integer…

Q: The Goldbach Conjecture says that every even integer greater than 3 is a sum of 2 primes. Prove the…

A: Given a statement The Goldbach Conjecture says that every even integer greater than 3 is a sum of 2…

Q: Show that 1+2+3+… +n = (";").

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Q: what

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Q: 2 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction…

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Q: (2n- 2. Give a combinatorial argument to show that k- ways the number of ways to select a committee…

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Q: 6"- (-4)" %3D k 1 k odd IM- 2]

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Q: then x* 1 is odd." Give a proof by contradicton to show "1fx s even integer,

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Q: ) Prove every integer n >= 7, n! > 3n

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Q: 6. Use contradiction to prove that, for all integers k > 1, 2/k+1+ 2/K+2

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Q: Το show 1² + 2² + 3² +4² +5² + To proove by induction; + n² n(n+1)(2n+1) 6

A: We have to show that 12+22+32+42+52+......+n2=nn+12n+16........1 by using principal of mathematical…

Q: Prove by contraposition that for all integers n, if n is odd, then 5n 4 is odd.

A: We need to prove if n is odd, then 5n - 4 is odd by the method of contraposition.  The…

Q: Prove that for all natural numbers k, a (5*2^(2k+1) - 1)/3 + a*4^k is of the form 5 + 6a.

A: The question looks incorrect and cannot be proved. This is because of the following.…

Q: Let m 2q be an even positive integer. Prove that every integer is congruent modulo m to a unique…

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Q: (c) (n² + n+ 41)² = 1 mod 12 , for every positive integer n

A: We know that,

Q: 13. Prove that 6 | (n³-n) for every integer n ≥ 0.

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Q: Prae that x4 -5x +3 is ineducible ouer Q

A: Given polynomial is f(x)= x^4 -5x^3 +3

Q: Write down the supremum of the set {7+ 7/n2 − 5/m | n,m ∈ N}. prove ur answer from the definition.

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Q: Prove that mn = nm holds for all m, n in natural numbers N. (Some lemmas will be needed.)

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Q: 13. Prove that 6 (n³-n) for every integer n ≥ 0.

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Q: III) Are there positive integers x and y suct that x³ = 2² +15? (Hint: for every integer x, x³ 0, 1,…

A: We shall show that no such positive integers exist.

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

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Q: Prove that for every positive integer n, there exist i and j with 1≤ i < j≤n +1 such that n is a…

A: Answer

Q: Show that 18! ≡ -1 (mod 437).

A: We have to show that: 18!≡-1mod437

Q: 17!=1(mod19). Select one: True False

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Q: Let a and b be odd positive integers. If there exist integers s and t with as+bt=4 then * O…

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Q: integer 107] (n²-71) ? Does there exist an such that

A: NO , DOES NOT EXIST SUCH INTEGER n

Q: Prove by cases that for all integers n, n +5n is even. Case 1 (n is even): Case 2 (n is odd)…

A: The given term is n2+5n

Q: Let 1 ≤k≤p-1 where p is a prime number. Then prove that (₁7¹) = (-1) (modp) k

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Q: Establish that if a is an odd integer, then 2" aʻ = 1 (mod 2" *2) for any n 2 1.

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Q: 6. Prove that 3" < n! whenever n is a positive integer greater than 6.

A: We need to prove that ,  3n &lt; n!  for every positive integer n greater than 6. We will use method…

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

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Q: Prove that 31 (n³ +5n+6) for every integer n ≥ 0.

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Q: 4) Let n 2 2 be an integer, such that 2" – 1 is prime number (e.g. if n = 7 then 2" – 1 = 127 is a…

A: Solution

Q: n + 5 > 4 n

A: We know, the square of an integer is always non-negative (n - 2)² ≥ 0 for all integers n

Q: "Proof 7: Suppose that a and b are both even. Then there exist integers k₁, k₂ such that a = 2k₁ and…

A: As per rule of bartleby we can solve only one question at a time and we can solve only three subpart…

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: If P(n) is the statement "n' + n is divisible by 3' Prove that P(3) is true but P(4) is not true.

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