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

Q: Prove by cases that 25x² + 35x is an even integer whenever 5x-7 is an integer. Note here that the…

A: Apply the condition for 5x-7 to be an integer case by case.

Q: Prove that 2n32n - 1 is always divisible by 17.

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Q: show that if x is an integer then x2+x-41= 0 produce prime numbers

A: Here show that if x is an integer then  produce prime numbers.

Q: 2. Prove that 11" – 7" is divisible by 4 for any positive integer п.

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Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: (a) Show that n(n2 – 1)(n + 2) is divisible by 4 for all integer n.

A: As per guidelines , if there are more than one questions , we have to solve the first question only…

Q: If 5|(2a + 1) for some integer a, then 25|(14a² + 19a – 19).

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Q: Show that for any integer n, n² – 2 is not divisible by 3.

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

Q: The product of two integers of the form 6k+5 is of the form 6k+1 Select one: True O False

A: 1. The word integer originated from the Latin word ''Integer'' which means whole. It is a special…

Q: Show by contraposition that for positive real numbers a and b, if ab - a - b ≥0 then a and b are…

A:

Q: If a, b, and c are integers such that b is a multiple of a³ and c is a multiple of b², then c is a…

A:

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: a

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

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Q: Show that for any integer n , n2 – 2 is not divisible by 3.

A:

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: Prove that if for some integers a, b, c we have that a° + 6° + c° is divisible by 9, thên at least…

A: Here we have to prove that if for some integers a,b,c we have that a3+b3+c3 is divisible by 9,then…

Q: Find the number of ordered pairs (m,n) such that m and n are positive integers in the set {1, 2,…

A: Let us consider Tm=2m+1.  m Tm 1 3 2 5 3 9 4 17 5 33 6 65 7 129 8 257

Q: Suppose, and no are the smallest of two positive integers so that 300x is a perfect square and 300y…

A: The given problem is to find the values of x and y and sum of value of x+y for which smallest…

Q: Prove that if for some integers a, b, c we have that a + b3 + c* is divisible by 9, then at least…

A:

Q: Find all possible values of P in x^2+10x+P=(x-A)(x-B) if P, A and B are all whole numbers

A: To find all possible values of P in x2+10x+P=(x-A)(x-B) if P,A and B are all whole numbers.

Q: find all possible three-digit numbers that are of the form 75_ and are divisible by both 3 and 2

A: Since the number is divisible by 2 and 3, it must be divisible by 6.

Q: Determine whether the statement is true or false. 1 dr In (8) x2 -

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

A: Consider the provided expression,

Q: (b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.

A:

Q: If a is divisible by b, then bq, for some integer q. las

A: If a is divisible by b, then, a = bq for some integer q.  TRUE

Q: SHOW THAT 3^2N - 1 IS DIVISIBLE BY 8 FOR ALL POSITIVE INTEGERS.

A: We prove this by induction.

Q: Prove that every integer greater than 27 can be written as 5a + 8b, where a, b e Z+.

A: Let P(n) be the statement, "For n∈ℕ there exist a, b>0 such that              n+27=a·5+b·8 Use…

Q: Find the term of (a – 2b)3 that has b in it. |

A:

Q: Develop a simple expression for the entries of A", where n is a positive integer and 1 A = -IN -IN

A: To get a simple expression for the entries of An, where n is a positive integer. A=112012

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: The set of all positive multiples of v2 is countable.

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Q: If a, b, and c are integers such that b is a multiple of a and e is a multiple of b², then c is a…

A:

Q: Let n = integer; 7 divides n if and only if 7 dividesn^2 (The result(s) ought to be integer(s))

A: Consider an integer n. We have to prove that 7 divides n if and only if 7 divides n2. Suppose that 7…

Q: Show that 3-4" + 51 is divisible by 3 and 9 for all positive integers

A:

Q: How many integer solutions to x1+ x2+ x3+ x4 = 36 are there for which æ1 > 5, x2 > 3, x3 > 5 and æ4…

A: We have to find the number of solutions to x1+x2+x3+x4=36 for which x1≥5, x2≥3, x3≥5 & x4≥6.

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

A:

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

A:

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

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

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Q: Find all x such that 7x+3 is divisible by 9.

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Q: 6. If x is any positive integer, which one of the following terms could be added to the sum of 2x +…

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Q: 2. 6n – 1 is divisible by 5 for n any positive integer

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Q: If a and b are odd positive integers, and the sum of the integers, less than a and greater than b,…

A: Let a and b are odd positive integers, and it is given that sum of integers, less than a and greater…

Q: Is it true that for any odd n the number 2" +1 is divisible by 3? Prove your answer.

A: We prove this by using PMI(Principle of Mathematical induction)

Q: Let A = {x | x is a integer and x^2 < 16}, identify each of the following as true or false:

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Q: 4 If x is an integer, then xr >x.

A: See the detailed solution below.

Q: If X is a positive integer divisible by 6, and X <60 What is the greatcst possibe vaive of X 7

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Q: For natural numbers k,m,n, and p, let k be the greatest common divisor of m and n, and let p<k. The…

A: Theorem: The greatest common divisor of m and n is the least positive value of an+bm where a and b…

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