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

Q: Let n ∈ Z. Prove that if n is divisible by 3, then n = 6k or n = 6k + 3 for some integer k.

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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 for a positive integer n we have that  33n+3 - 26n - 27 is divisible by 169.

A: To show  33n+3 - 26n - 27 is divisible by 169.

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 or disprove: For every positive integer n, 22n – 1 is divisible by 3. -

A: I have proved the result using mathematical induction

Q: A. Prove that for all positive natural numbers n, 13 + 23 + . + n³ = n²-(n+1)²/4. + ...

A: Explanation of the answer is as follows

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

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Q: Q2 Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

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

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

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

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: For any non negative integer n , prove that an − bn is divisible by a−b,where a and b are any…

A: answer

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

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Q: Find all integers n > 1 such that 1n − 1)" is +2+...+(n divisible by n.

A: To find all the integers n>1 such that 1n+2n+...+n-1n is divisible by n

Q: Let n be a positive integer. Prove that (:) () (:) 1. 1 + 2. 2 +n n2"-1

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Q: Prove that if  n  is a positive integer, then  5 | (6^n − 1)

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Q: Use the Principie to prove the Ving 3n+1 – 1 is divisible by 2 for all n > 0

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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: Use induction to prove that for every integer 15"+2 + 162n+1 is divisible by 241. n > 0,

A: To use mathematical induction to prove the given statement

Q: Prove that for all positive integers n, 4n-1 is divisible by 3.

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

A: We have to prove the following -  The fourth power of any integer has the form 4k or 4k+1 for some…

Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: Use mathematical induction to prove that 3n+ 7n−2 is divisible by 8 for every non-negative integer…

A: The given statement is, Pn=3n+7n-2 Let us assume that the statement is true for n=1 P1=31+71-2=8 As…

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: Use Induction to prove that 22n+1 + 32n+1 is divisible by 5 for all n > 0.

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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 that2.7n + 3.5n – 5 is divisible by 24, for all n ∈ N.

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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: Suppose n is an integer. Prove that if n is even, then n²+3n+5 is odd.

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Q: Prove, using any method, the following statement: if n is an integer, then (3n+1)(5n+4) is an even…

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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: Use induction on n to prove that 42n – 15n – 1 is divisible by 225 for all non-negative integers.

A: Prove 42n-15n-1 is divisible by 225 for all non-negative integers using mathematical induction.

Q: Prove that n2-n is divisible by 2 for every positive integer n.

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Q: Use the PMI to prove that 6" – 1 is divisible by 5 for every integer n > 0.

A: To prove that 6^n-1 is divisible by 5 for every integer greater that or equals to 0.

Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

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Q: Use induction to prove that for every integer n  ≥ 0, 5n+2+ 62n+1 is divisible by 31.

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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: Prove that 8" – 14n + 27 is divisible by 7, for all positive integers n.

A: The objective is to prove 8^n-14n+27 is divisible by 7.

Q: Prove, using mathematical induction, that 8n−3n is divisible by 5 for all integers n≥0.

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

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Q: Let suppose integers are randomly selected. Prove that the difference of two of them is divisible by…

A: Solution

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: Use induction to prove that n^3 + 2n is divisible by three for all integers n greater than or equal…

A: Given information:  n3+2n is divisible by three for all integers n greater than or equal to zero.…

Q: For any non negative integer n , prove that an – bn is divisible by a-b,where a and b are any…

A: Use the binomial factorisation