Prove that if n is an odd integer, then n3 +n+1 is also an odd integer.

Q: Prove that for every integer n≥0, the number 42n+1+3n+2 is a multiple of 13.

A: Let us prove the given statement by mathematical induction. Case 1: Let n=0

Q: Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

A: n is an integer. To show: 2n3 + 4 is divisible by 4 then n is even.

Q: Prove: For integers m, n, 12mn – 9 is an odd integer.

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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: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

Q: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

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

A: To prove: 1+4+7+...+3n-2=n3n-12

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

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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 that for any natural number N, there exists N consecutive integers none of which is a power of…

A: Please see the attachment

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: Prove that n²– 3n + 1 is odd for every integer n.

A: To prove that n^2-3n+1 is odd for every integer n.

Q: Prove the following statement by contradiction: If (n – 3)2 is an odd integer, then n is an even…

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Q: Let n be a positive integer. Prove that (:) () (:) 1. 1 + 2. 2 +n n2"-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 if  n  is a positive integer, then  5 | (6^n − 1)

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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 n2-n is divisible by 2 for every positive integer n

A: We have to prove that,  n2-n is divisible by 2 for every positive integer n. We will use…

Q: Prove that if n is an integer and 5n + 6 is even, then n is even.

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

A: we have to prove that the statement is true for every positive integer n.…

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

A: we have to prove that the statement is true for every positive integer n. 4 is a factor of 5n-1

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 if n is an integer and n² is an even integer, then n is an even integer

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Q: Let n be a positive integer. Prove that if n2 is even, so is n.

A: We have,  n be a positive integer and an even number. Therefore, n=2k; k∈ℝ

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: Show that there are infinitely many integers n such that 4n^2+1 is simultaneously divisible by 13…

A: We need to prove there are infinitely many integers n such that 4n^2+1 issimultaneously divisible by…

Q: Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contraposition

A: Given , n is an integer .To prove → if 3n+2 is even , then n is even using a proof by composition…

Q: Prove that for any integer n, n' -n is even.

A: CASE I When n is even Let n=2m,m∈N⇒n3−n=2m3−2m⇒2m4m2−1=2η,where η=m4m2−1Thus,n3−n is even when n is…

Q: prove that if a and b are both odd integers, then ab+1 is an even integer

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Q: Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even.

A: (if part) : Let, n be an even positive integer. Then, n = 2k for some positive integer k. So, 7n+4 =…

Q: prove that Every positive integer n can be written as n =( 2^a)(5^b)c, where c is not divisible by 2…

A: Fundamental Theorem of Arithmetic: Every positive integer (except n=1) can be uniquely expressed as…

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: Prove that a positive integer n is p

A: Topic :- Number system

Q: Prove that 3, 5, and 7 are the only three consecutive odd integersthat are prime.

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Q: Prove by any means that n!/(n − r)!r! is always an integer for 0 ≤ r ≤ n

A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

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 n2-n is divisible by 2 for every positive integer n.

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Q: Prove that the statements here are true for every positive integer n. 2 + 22 + 23 +……++ 2n = 2(2n -…

A: The given statement is: 2+22+23+...+2n=2(2n-1) We need to prove that the given statement is true for…

Q: Prove that the statements here are true for every positive integer n. 12 + 22 + 32 + · · · + n2=n(n…

A: Proved that for n≥1 fn=12+22+32+⋯⋯+n2=nn+12n+16 For n=1 LHS=12=1 RHS=nn+12n+16=11+12+16=66=1…

Q: Prove that for any odd positive integer, n, the sum of n consecutive positive integers is divisible…

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Q: Prove that the statements here are true for every positive integer n. 1 + 3 + 5 + · · · + ( 2n - 1)…

A: Show that for every positive integer n    1 + 3 + 5 + · · · + ( 2n - 1) = n2

Q: Show that for every integer n ≥ 8, there exist integers a ≥ 0 and b ≥ 0 such that n = 3a + 5b.

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Q: Show that if n is an integer, thenn= [n/2] + [n/2].

A: Floor function x: x gives as output the greatest integer less than or equal to x. Ceiling function…

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

A: 13+23+33+---+n3=nn+122 Let us use the Principle of Mathematical Induction to prove the given…

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: Prove that n2 − 3n + 1 is odd for every integer n.

A: Prove the result using mathematical induction. P(n):n2-3n+1. Check whether the result is true for n…

Q: Prove that for any integer n, if n³ is odd then n is odd.

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Q: Prove by contradiction that 6n + 5 is odd for all integers n.

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Prove that the statements here are true for every positive integer n. 13 + 33 + 53 + · · · + (2n -…

A: Prove that the statements here are true for every positive integer n. 13 + 33 + 53 + · · · + (2n -…

Q: Show that for all positive integers n and k, (~^)= (-1)^(" + k-1). k

A: The provided information is  (  k-n) Then we prove this (   k-n)=(-n)*(-n-1)*(-n-2)*.....(-n-k+1)k!…