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

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: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

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

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Q: 5. Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers.

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Q: Provide a proof by contradiction of the statement below. For all integers x, 3x + 2 is not divisible…

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Q: 8.) Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers. 9.) Is 9"…

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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: Use the definition of even to verify that the expression is an even integer. 12m + 4n+10 2-10m

A: An even integer is an integer which is divisible by 2. Let a is an even integer i.e. a is divisible…

Q: Prove, using the definition of odd, that for all integers r, that 6r+5 is odd.  b) true or false:…

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Q: Prove that a number in the form 6k+5,” what I mean is a number in the form of 6 times some integer…

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Q: (2) Show that for any integers a,b we always have a? – 4b # 2. -

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Q: Use indirect proof to show that, "If n is an integer and 3n+2 is even, 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: (a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer…

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Q: Use a direct proof to show that every odd integer is a difference of two squares. (For example: 1 =…

A: first we take a odd integer then we write in 2y+1 form then we transfer to difference of two…

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: divisible by 3 for

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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: Prove that the statement is true for every positive integer n.

A: We know that the sum of a geometric series of the form: a+ar+ar2+ar3+...........+arn-1 with the…

Q: Express the following statement using N-notation.: 12.5 (3x + 4) for all real numbers x > 2. x + 2

A: In the given question the objective is to express the following statement using notation

Q: Use contradiction to prove:  if 3n + 2 is an odd integer, then n is odd.

A: consider 3n+2 as odd integer  and let  n is even  therefore, n=2k for some integer k therefore,…

Q: True or False: There are integers x, y such that 15 = 215x + 497y. O True O False

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Q: Prove the statement for positive integer n and r, with r ≤ n. ( n n − r ) = ( n r )

A: We know that, the term nr for positive integer n and r, with r ≤ n is given as nr=Crnnr=n!n-r!r!…

Q: Which of the following statements is (are) true for all positive integers? 1) n' + 2n is divisible…

A: given statement 1)  n3+2n  is divisible by 32) 52n-1  is divisible by 24

Q: se indirect proof to show that, "If n is an integer and 3n + 2 is even, then n is even

A: Let us solve the classical algebra problem in the next step.

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: 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: Let a be an integer such that the remainder of dividing a by 17 is 15. What is the remainder of…

A: We will find out the required value.

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: Use proof by contradiction to show that for all integers m, 7m+4 is not divisible by 7.

A: Recall the following fact about divisibility.

Q: 2. Determine all values of x - y when x and y are integers such that x = 360 + y².

A: The given equation is. x2=360+y2x2-y2=360x+yx-y=360 Since it is given that x and y are integers, so…

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

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

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

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

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Q: Prove that 3" + 1 is divisible by 4 whenever n is an odd positive integer.

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Q: Show for all integers n2 0 that 91 (4" - 1). a.

A: A. We will solve the question using the method of mathematical induction.

Q: (a) Show that 1++++ 1 21+, 2n-1 2n whenever n is a nonnegative integer.

A: We need to show that 1+12 + 13 + ..... + 12n-1 + 12n ≥ 1+n2We will show this by mathematical…

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

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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: 1. Prove: 11" – 6 is divisible by 5 for all positive integers, n.

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Q: 2 ) + ("**) = n²- .2 2

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Q: For all integers a, b and c, if a divides b and a divides c, then a divides (4b - 3c). True False

A: Answer is TRUE

Q: Using the direct proof method, prove that for every integer a, if (a-2) is divisible by 3, then…

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Q: a) Find the number of positive integers not exceeding 120 that are divisible by 3 or 4. 1 0 01 0 0 0…

A: As per our guides lines I answered only 1st part