Which of the following statements is (are) true for all positive integers? 1) n' + 2n is divisible by 3. 2) 5* – 1 is divisible by 24.

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: Prove that the analogue of lemma 1.23 is not true for numbers of the form 4n+3 where n is an…

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Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: 4. Use the PMI to prove the following for all natural numbers n. |(c) E-1 2' 2n+1 – 2 i=D1

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Q: Prove that if n is an integer and 7n+2 is even, then n is even. State which method you are using for…

A: Let  n be any integer.  We have given that , 7n + 2 is even. We need to prove that ,  n is even. We…

Q: Prove that the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n.

Q: 5. Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers.

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Q: 8.) Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers. 9.) Is 9"…

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Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

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Q: Show that 3n^ 2+8nlogn is big-Oh (n ^2 )

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Q: 4. Prove that given any integer for n, n3 + 2n will be divisible by 3. 5. Write the formal proof…

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Q: Prove or disprove that for all n > 1 п i2' < 2"+1n. i=1

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Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

Q: prove that n(13n²-1) is divisible by 6 for every n ≥1

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Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

Q: 7. For n > 1, verify that 1? + 3? + 5² + ...+ (2n – 1)? = ( 2" *1 ) %3D -

A: We have to prove that for n&gt;=1 in question number (7) Sum of squares of first n natural numbers…

Q: 2. Prove by contraposition: For all integers n, if (3n +4) is even, then n is even.

A: Contrapositive of 'if p then q' is if not q the not p'

Q: 7) Use a direct proof to show that: a) The sum of an even integer and an odd integer is odd. b) If n…

A: Use the following concepts, to obtain the required result. An even integer m can be written as a…

Q: Prove by cases that n2−1 is divisible by 3 when n is an integer not divisible by 3.

A: The detailed solution is given below Thank you

Q: 3. Find integers s and t so that 1 7s + 11t. Show that they are not unique.

A: The division algorithm states that for any integer, a, and any positive integer, b, there exists…

Q: (b) Prove by induction for all n >1 that an – bn = 2bn-1· %3D

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Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

Q: Prove that lor n {k=1 positive unteger every K2K = (n-1)2²+¹+2

A: answer is in next step.

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

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Q: 17. 1+3n 0.

A: Alogrithm: Obtained P(n) and understand it's meaning. Prove that statement P(1) is true. Assume…

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: (d) Prove by induction that 5 "– 1 is divisible by 5 for all n e Z +.

A: we have to show that 5n-1 is divisible by 5 for all n belongs to Z+ by induction for n =1.

Q: nº +n² + 1 4. Prove that = O(n³)

A: To prove: n4+n2+1n=Θn3

Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

Q: 3. By using mathematical induction proves 2' + 22 + 2 +.. +2" = 2"*! – 2 is true for all positive 3…

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Q: Use a suitable method to prove or disprove the following statement formally and precisely: If the…

A: Even and odd numbers can also be characterized as: Even Number: A number that is divisible by 2,…

Q: 2. Prove that if n is even, then 3n + 1 is odd

A: The given expression is 3n + 1. Prove that 3n + 1 is odd, if n is even.

Q: Verify by mathematical induction Ž(2i-1 1. 1 - n° (2n? – 1) for all n 2 1. n(n+1) 2. Prove: V n € Z,

A: As per our guidelines, we are allowed to answer only one question. Since you have asked multiple…

Q: II. Prove the following using Direct Proof. 6. If x is an odd number and y is even number, then x +…

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Q: Determine whether 5 and 7 are irreducible or not in Zi]. O 5 is reducible but 7 is irreducible. O 5…

A: We have to check

Q: 2) If n is an odd integer, then H= (n-1) 2

A: To prove that n2=n-12 such that n is an odd integer.

Q: 3. Find integers s and t so that 1 7s+ 11t. Show that they are not unique. %3D

A: Apply the division algorithm, we get 11 = 7*1 + 4 7 = 4*1 + 3 4 = 3*1 + 1 Hence, 1 = 4 - 3*1   = 4 -…

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

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Q: Determine whether 5 and 7 are irreducible or not in Zi. O 5 is reducible but 7 is irreducible. O 5…

A: Here , both 5 and 7 are irreducible because both cannot be reduced further. As both are neither in…

Q: 1. Prove by the Principle of Mathematical Induction that 1 x 1! + 2 x 2! + 3 x 3! + ... +nxn! = (n +…

A: see below the explanation

Q: Suppose n E Z. Prove that if n is even, then n² + 7n + 10 is even. (Use Direct Proof)

A: Suppose that n∈Z and let n is even.To show: n2+7n+10 is also even.

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

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Q: 1. Prove that if 7n2 + 5 is even, then n is odd

A: The given statement is,

Q: 11. If n is an odd integer, show that n + 4n² + 11 is of the form 16k.

A: If n is an an odd integer then we have to show that n4 + 4n2 + 11 is of the form 16k for some…

Q: 1.Then for positive integer n, A" is 0 1 Let A = 0 n² In 1 II In 0 1 II 0 0 2n In-1 n-1 2.

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Q: 6. Prove that for n> 2, 2(;) + (*) =n². п

A: We have too prove the given expression.   According to our guidelines we can solve only one question…

Q: give direct proof by contradiction that is 5n-1 is odd, then n is even

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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…

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