Prove that the fourth power of any integer has the form 4k or 4k + 1 for some integer k.

Q: Show that every positive integer is of the form 2q,and that every positive odd integer is of the…

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Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

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

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

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: Prove: there exist no integer a and b for which 21a + 30b = 1.

A: We can prove this by contradiction. Take the negation of the statement and check if it is true, if…

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

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Q: For any integer k, prove that gcd(2k +3, 5k +7) = 1.

A: Use the given information.

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 the statement is true for every positive integer n.

A: 11·2+12·3+13·4+---+1nn+1=nn+1 Let us prove this statement by using the Principle of Mathematical…

Q: 2. Prove that the product of any two odd integers is an odd integer.

A: Let x and y be two odd numbers. If a and b are odd, then a and b can be written as: x=2m+1 , y=2n+1…

Q: Prove 1 +,+ i++ = 2 – for all positive integers n. |

A: We want to prove 1 + 12 + 14 + ... + 12n = 2 - 12n

Q: For every positive integer n, prove that 7n – 3n is divisible by 4.

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Q: Prove the square of any odd integer is expressible in the form 8n+1 with n ∈Z.

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Q: Prove that 5n-1 is divisible by 4 to any positive integer n

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

A: Topic - principle of mathematical induction

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: Prove: for every positive integer n, 2"> n

A: As per the guidelines, since there are more than one question only one question will be answered.…

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: Suppose a is an integer. Prove that if a^3 is a multiple of 3, then a is a multiple of 3.

A: Let a3 is a multiple of 3 but a is not a multiple of 3.a3 is a multiple of 3Therfore, 3 divides…

Q: Prove by mathematical induction that 3n+2n+3 is divisible by 4 for all positive integers

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Q: Show that the cube for any integer is of the form 7k, or 7k ± 1

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Q: Prove that for all natural numbers n, (n^3/3) + (n^5/5) + (7n/15) is an integer

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

A: Given,  2+7+12+...+(5n-3)=12n(5n-1)

Q: Prove that a positive integer a > 1 is a square if and only if in the canonical form of a all the…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: Let k be some positive integer. Prove that a number of the form 4k +3 can never be the sum of two…

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

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Q: Prove that there exists a positive integer n > 2 such that 4n+1 is a prime number.

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Q: 85. (a) Prove that any integer is expressible in the form 4n, 4n + 1, 4n + 2, or 4n + 3 where n is…

A: Here we use mainly dividend = divisor × quotient + remainder.

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: Prove that the statement is true for every positive integer n.3 is a factor of n3 + 2n

A: Consider the given statement, "3 is a factor of  n3+2n" for all positive integer n Suppose Pn…

Q: Prove that there is no integer n for such that 5n+3 is divisible by 5.

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Q: Prove that if n is an integer and n² is an even integer, then n is an even integer

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Q: Prove that the fourth power of any integer has one of the forms 5k or5k + 1, where k is an integer.

A: We need to prove that any integer has one of the forms 5k or 5k + 1, where k is an integer.Let p be…

Q: Prove that I +:+ + for every positive integer n.

A: In this question, principle of mathematical induction is used. Principle of Mathematical Induction…

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

A: We have, Pn=a2n-1+b2n-1 Let us consider the statement Pn is true for ∀n∈ℕ. Therefore,…

Q: Prove that every prime greater than 3 can be written in the form 6n + 1 or 6n + 5.

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

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

A: Given, Prove this statement is true for every positive integer n: 2+4+6+..+2n=n(n+1)

Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

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

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Q: prove that there is no positive integer n such that n^2+n# = 100

A: n^2+n^3 = 100 We can evaluate this till some positive integers

Q: Show that any integer of the form 6k + 5 is also of the form 3j + 2, but not conversely.

A: 2. We can write 6k+5 = 6k+3+2=3(2k+1)+2 lets write 2k+1=j hence 6k+5=3j+2

Q: A. Prove that n3 + 2n is divisible by 3 for all integers n.

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Q: Prove that 3" >n² for all positive integers

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Q: Prove that if n is an integer, then 2n2 + n + 1 is not divisible by

A: The given problem is to prove the following statement using the possibile cases. We need to find the…

Q: Let n be any integer. Using a proof by cases, prove that n4 is either in the form of 5k or 5k+1 for…

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

A: a>1, then an>1

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.