Prove that If a and y are integers and r + y is odd , then a – y is odd. I Show that for any integer n , n² + 2 is not divisible by 4.

Q: Prove that 6 divides n^3 − n whenever n is a non-negative integer

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Q: show that n(n^2-1)(n+2) is divisible by 4 for all integers n

A: To show that n(n^2-1)(n+2) is divisible by 4 for all integers n

Q: Prove by cases For any integer n, n^2 + 5 is not divisible

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Q: Jse a direct proof to show that if n² is an odd integer, then n* is also an odd integer.

A: Given, n² is an odd integer. So, n² = 2k+1, where k is an integer. Now, n⁴ = (n²)² = (2k+1)² = 4k² +…

Q: show that every integer greater than 11 is the sum of two composite integers

A: We need to prove that  every integer greater than 11 is the sum of two composite integers.

Q: prove the difference of the squares of two z integers is always divisible by 16 (i.e. z1^2 - z2^2 =…

A: We prove that (z_1)^2-(z_2)^2=16k for some integer k, then we say that divisible by 16.

Q: Prove that a four-digit number is a multiple of 8 whenever the number formed by the first three…

A: The solution is given by using the expanded form of number as follows

Q: Prove that (-1+ iB)" + (-1- i3)" has either the velue 2t! or the value if a B any integer (positive,…

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Q: The representation of an even integer is 2k; and that for odd is 2k+1 or 2k-1 for some integer k.…

A: Given the statements: The representation of an even integer is 2k for some integer k. The…

Q: Show that a6-1 is divisible by 168 whenever (a, 42)=1.

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Q: Prove that 3a^1 is even if and only if a is an odd integer.

A: In the given question we have to prove that Prove 3a^1 is even if and only if a is an odd integer.

Q: Prove that n-5 is odd iff n^2-1 is odd

A: Please check step 2 for the solution.

Q: Prove that 30 divides n° – n for every nonnegative integer n. -

A: The given question can be solved as shown in step2.

Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

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