5. Prove that for any integer n, at least one of the integers n, n+2, n +4 is divisible by 3.

Q: Let n be an integer. Prove that GCD(n² n + 1, n + 1) equals 1 or 3. Show by examples that both…

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Q: For any integer z, if 3 divides z2, then 3 also divides z.

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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…

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Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

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Q: Prove that 1+2+3+4+...+n= n² + n 2 for every positive integer

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Q: Determine whether the following statement is true or false. Justify your answer with a proof or a…

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Q: Compute for an arbitrary integer k

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Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

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Q: 3. Let a and b be positive integers. Express the positive greatest common divisor of a and b in…

A: The positive greatest common divisor (gcd) of two positive integers, a and b, can be expressed in…

Q: Prove that for any integer n, we have n - 2 is not divisible by 4.

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Q: Prove that, all integers x and y, x^2-4y≠2  1a. Prove that all numbers of the form 1007, 10017,…

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Q: Find all integers x + 3 such that x – 3 is divisible by x – 3. |

A: To find all integers such that x^3-3 is divisible by x-3

Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n>1. All prime divisors of n!+1 must be bigger than n. This is…

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

A: To prove for any odd positive integer, n, the sum of nconsecutive positive integers is divisible by…

Q: divisible by 3 for

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Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: We have to prove that and  are relatively prime for every integer .We know thatThe greatest common…

Q: Show that for every positive

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Q: Suppose that 75% of students attend lectures for a particular course and 20% of students fall the…

A: Conditional probability : Conditional probability is a measure  of the probability of an event…

Q: Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder…

A: we have to Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the…

Q: 8) Prove that n³ + 2n is divisible by 3 for any natural n. 5

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Q: if a is a positive integer less than 10, then 7a7,7a7,7a7 is divisible by 3 Select one: O True O…

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Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

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

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Q: For all integers n> 0, 32n – 1 is divisible by 8.

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Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…

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

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Q: 3. Use induction to prove that 15"+2 + 162n+1 is divisible by 241 for every nonnegative integer п.

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Q: prove that n²_1 is divisible by when ever n is odd an Positive integer.

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Q: prove that it ņ is an 3n +2 is odd then integer and is add

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Q: 10. Prove that n³-n is divisible by 6 for every integer n.

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Q: Show that if 2n−1 is prime then n is prime.

A: We have to show that if 2n-1 is prime then n is prime.