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

Q: If x is any positive integer , then the sum of 6x and 11x is always divisible by what?

A: x is a positive integer. Find sum of 6x and 11x is divisible by what?

Q: Which of the following rules is used to find the number of positive integers less than 1000 that are…

A: To Determine: which of the following rules is used to find the number of positive integers less than…

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

A: In the given question n>1, there arises two cases as follows- Case 1- When n is odd that is…

Q: 4) Prove that the product of any three consecutive integers is divisible by 6.

A: We have to show that product of any three consecutive integers is divisible by 6

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

A: Even number: we say a number 'n' is even if it is divisible by 2. if  'n' is even then it is…

Q: Show that an integer N is divisible by 9 if and only if the sum of its digits is divisible by 9

A: Let a=anan-1...a3a2a1 be an integer and S be the sum of the digits. Consider a,…

Q: Show that every positive integer can be written as the product of a square (possibly 1) and a…

A:

Q: Find the product of all positive integers from n down through 1 for n = 5.

A: Concept: The calculus helps in understanding the changes between values that are related by a…

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: what two consective odd integers have a product of 63?

A: Find two odd integers.

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

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

A: Given , 3.4n+51 is divisible by 3 and 9 for all positive integers n''-1The statement 1 can be…

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: For each pair of integers r and s below, Find the greatest common divisor d. Then, find m, n, such…

A: To determine the greatest common divisor, d and then to find m and n such that d = mr + ns. (a) r=32…

Q: AV Prove that every integer greater than 6 can be written as 3x + 4y for nonnegative integers x and…

A: We have to prove that:Every integer greater than 6 can be written as 3x+4y for nonnegative integers…

Q: Prove that 13n13 +11n is always divisible by 24.

A: We prove this by induction on n.

Q: Find the prime factorization of (2) and of 20!. 10

A: To find the prime factorization of 2010 and of 20!. Prime factorization means we have to represent…

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

A:

Q: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

A:

Q: Find all positive integers n for which n1º +1 is divisible by 10.

A:

Q: Use proof by contradiction to show that the sum of two odd integers(k and l) is eve

A: An even number is a number that is divisible by two. All even numbers finish with one of the digits…

Q: prove n2 + 5 greater than or equal to 4n for all integers n

A:

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

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

Q: DO THIS TYPEWRITTEN FOR UPVOTE

A: Given that the representation of an even integer is  and that for odd is  or  for some integer .The…

Q: Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4. In…

A: Introduction: Divisibility is a property of an integer. For two integers a and b with b>a, the…