. For a positive integer a, determine whether the following statements are alw times true, or never true. (a) is greater than 2 (b) is greater than .

Q: Prove that if n is an odd integer, then n3 +n+1 is also an odd integer.

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Q: Any integer of the form 12k + 11 is also of the form а) Зт + 1 b) 5m + 2 4т + 3 бт + 1

A: Step:-1 Given integer form is 12 k +11 Take  a =12 k +11  then a=12 k +11⇒a=12 k +8+3⇒a=43k +2…

Q: Prove that the cube of any integer has to be exactly one of these forms: 9k, 9k + 1, or 9k + 8 for…

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Q: -. For any integer not an integer, then [x]+[m- x] = m-1. m and any real number x, if x is

A: Let m be any integer and x be any real number that is not an integer.

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

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

Q: Given n < 10" for a fixed positive integer n°2, prove that (n + 1)* < 10*1.

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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: If is an odd integer, show that n* + 4n2 + 11 is of the form 16k.

A: Firstly, we have to prove that n4+4n2+11 is in the of 16k for an odd integer n. for making the…

Q: If we want to prove by contradiction the statement: For any positive real number x there exists a…

A: We have to choose the correct answer that we should assume to prove the given statement by…

Q: Any integer of the form 12k + 11 is also of the form а) Зт + 1 b) 5m + 2 4m + 3 бт + 1

A: We have to do proper manipulation.

Q: -. For any integer m and any real number x, if x is not an integer, then [x]+[m- x] = m-1.

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Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

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

A: To prove: 1+4+7+...+3n-2=n3n-12

Q: Prove that the sum of the first n cubes is equal to the square of the sum of the first n positive…

A: According to the given information, it is required to prove the following.

Q: If we want to prove by contradiction the statement: For any positive real number x there exists a…

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Q: Prove by contradiction : For any odd integer n : n² – 7 is not divisible by 4.

A: Proving a statement by contradiction means assume the negation of statement true and prove the…

Q: Show that a fraction r = a/ b in lowest terms has a finite decimalexpansion if and only ifb = 2n5m…

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Q: 3. Let x be an integer. Use the previous result to prove the following corollaries. (a) If x divides…

A: Divisibility: If a and b are integers, we say that a divides b, and write a|b, if there is an…

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

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

Q: If we want to prove by contradiction the statement: For any positive real number x there exists a…

A:

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: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

Q: Verify that each of the following statements is a false statement by finding a counterexample for…

A: [a]n2&gt;nput n=002&gt;00&gt;0which is not trueput n=112&gt;11&gt;1which is not truetherefore,for…

Q: b) Prove that () - 2) = n², where n is a positive integer.

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Q: For every pair of integers x and y, if x – y is odd, then x is odd or y is odd.

A: Let x-is even integer and y-is even integer. x=2m, y=2n

Q: Prove that the fourth power of any integer has the form 4k or 4k + 1 for some integer k.

A: We have to prove the following -  The fourth power of any integer has the form 4k or 4k+1 for some…

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

A: we have to prove that the statement is true for every positive integer n.…

Q: Prove that 3" > n2 for every positive integer n.

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Q: Give a direct proof of the following statement. If x is an even integer, then x² – 6x + 5 is odd. |

A: Remark: We know that addition or subtraction of an even and an odd integer is always an odd integer.…

Q: R.4. Prove that for every integer n, n³ is even if and only if n is even.

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Q: or disprove : [nx] =n [x] %3D for all Positive integers n > 2

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Q: If a and b are distinct integers prove that a-b is a factor of an -b" whenever n is positive integer

A: if a and b are distinct integers prove that a-b is a factor of a^n - b^n whenever n is positive…

Q: Let A = {n E Z n = 5r for some integer r} and = {m e Z | m = 20s for some integer s}. Prove or…

A: We need ton prove or disprove given statement. We will prove in general and To Disprove we need only…

Q: 2) Prove that each of the following statements is false by giving a counterexample. a) If x is even,…

A: We have to check

Q: Prove that the statements here are true for every positive integer n. a + (a + d) + (a + 2d) + · · ·…

A: To Determine: Prove that the statements here are true for every positive integer n. Given: we have…

Q: Verify that each of the following statements is a false statement by finding a counterexample for…

A: Here given n2&gt;n for real number n Now put n=0 Thus statement becomes  0&lt;0 which is impossible.…

Q: 1. Prove the following using direct proof. a. If 5x – 3 = x + 5, then x = 2. b. If x and y are both…

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Q: Prove that if the product of two integers qs is odd, then q2 + s2 is even. Then show that for…

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Q: For every integer n23 I+ 2+ 2?+ +4^= 4(4n-16) %3D 3. Proof (by induction)

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

A: Given, Prove this statement is true for every positive integer n: 2+4+6+..+2n=n(n+1)

Q: 4. Let A = {nEZ|n= 5r for some integer r} and B = {m E Z|m = 20s for some integer s}. Prove or…

A: The given sets are as follows.  A=n∈ℤ:n=5r, r∈ℤB=m∈ℤ:n=20s, s∈ℤ a. The elements in the set A are the…

Q: prove the following using either direct or indirect proof  a. if a is an odd integer, then…

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Q: Prove that the statements here are true for every positive integer n. 1 + 3 + 5 + · · · + ( 2n - 1)…

A: Show that for every positive integer n    1 + 3 + 5 + · · · + ( 2n - 1) = n2

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

A: Given : n is an integer (3n+2) is an odd integer To prove : n is an odd integer

Q: 1. Ifx is an even integer, then x is even.

A:

Q: If n is an odd integer, show that n4 + 4n2 + 11 is of the form 16k.

A: Solution: The objective is to show that given equation is of the form 16k

Q: Prove that if n is an odd integer, then 3n+ 5 is an even integer.

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Q: a) Prove, for all integers k, that 2k-1 is odd. b)prove, for all integers x, if x is odd, then x =…

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Q: Verify that each of the following statements is a false statement by Finding a counterexample for…

A: Given: Statement: a. n2&gt;n b. nn=1

Q: Use the factor theorem to verify the statement

A: Given statement is, x+y is a factor of xn-yn for every positive even integer n.   So, Let fx=xn-yn,…