Prove that if a is an integer and b is a positive integer, then exists unique integers q and r with −b/2 < r ≤ b/2 such that a = qb + r.

Q: Show that every positive integer is of the form 2q,and that every positive odd integer is of the…

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Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

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Q: Prove that for all positive integers k and n, with k < n, n-1) k-1/ k 1 k ... k k- 1 k- 1

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Q: 3. Prove that for all x > 0 and all positive integers n, x3 et >1+x + 2! 3! n!

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Q: Let n be any integer. Prove that 3n - 2 is odd if and only if n is odd. Suppose x and y are any…

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Q: Prove that for all integers m>0 , m^2+5m+4 is composite.

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Q: Suppose x and y are any integers. Prove that if x and y are odd, then x+ 5y² is even

A: Suppose x and y are integers. Prove that if x and y are odd, then x+5y2 is even.

Q: Prove that for all integers a,b which are not both 0, and for all integers d, d|gcd(a,b) if and only…

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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: Let x and y be integers. Prove that if x2 + y2 is odd, then x+y is odd

A: Given x and y be an integers. We have to prove that if x2+y2 is odd then x+y is odd. Since given…

Q: Let x and y be integers. Prove that if x2 - y2 is even, then x + y is odd

A: The given statement looks wrong. If two numbers x and y are integers and x2-y2 is even  Now the…

Q: Let x,y,z∈ Z be integers. Prove that if x(y+z) is odd, then x is odd and at least one of y or z is…

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Q: Suppose if p and q are odd integers with p!= q (p is not equal to q). Show that there is a unique…

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Q: Prove that if m is a positive integer and r is a real number, then Lma) = l=] + -+ + + [-+ + r+ m

A: We are given a Inequality involving the sum of floor functions, and we have to prove that.

Q: integer of the form okts is also but not' conversely

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Q: Prove that for every integer x, x+4 is odd if and only if x+7 is even

A: An even number is defined as an integer of the form n = 2k, where k is an integer. An odd number is…

Q: There exists an integer x>0, such that for all integers y>0, there exists an integers z>0, so that…

A: We have to find out

Q: Prove, for all integers x, that if x2 is odd, then x is odd.

A: Result: Every odd integer n can be written as n=2k+1, where k is an integer. Every even integer n…

Q: Prove that for every positive integer n, 1 1+ V2 1 +...+ V3 1 > 2(Vn+1– 1) Vn

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Q: Prove that if n is an integer and n² is an even integer, then n is an even integer

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Q: Prove that I +:+ + for every positive integer n.

A: In this question, principle of mathematical induction is used. Principle of Mathematical Induction…

Q: Let k and n be two positive integers, and assume that n is odd. Prove that there exists an integer a…

A: Given: Let k and n be two positive integers in that, we assume if n is odd.

Q: Prove that there are no integers m and n such that m2=4n+2.

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Q: Prove that for any positive integers a₁, a2,..., an, there exists integers 1, 2,...,n, such that a₁₁…

A: To Prove: for any integers a1,a2,...,an there exists integers x1,x2,...,xn, such that…

Q: Prove that if x is an even integer, thenx^2−6x+ 3is odd

A: Any integer of the form 2m+1 is odd.

Q: a. Prove that for every integer a, if a³ is even then a is even. b. Prove that 2 is irrational.

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Q: Let a and b be odd positive integers. If there exist integers s and t with as+bt=4 then * O…

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Q: . Use direct proof to prove that, if x is a positive odd integer, then z – 1 is wholly dividable у…

A: x^2-1 is divisible by 4.

Q: Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even.

A: (if part) : Let, n be an even positive integer. Then, n = 2k for some positive integer k. So, 7n+4 =…

Q: Prove that for every integer n 2 0, gcd(Fn + 1, Fn) = 1.

A: Given: Fn is a Fibonacci sequence, with F0=1=F1. To prove: For every integer n≥0, gcdFn+1, Fn=1.

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: Prove that if ∣A∣ = 1 and all entries of A are integers, then all entries of ∣A−1∣ must also be…

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Q: Prove that the difference between two integers is odd if and only if exactly one of the integers is…

A: Suppose x be any integer. If x is odd then x=2m+1 for some integer m. If x is even then x=2m for…

Q: Recall that a is a type 1 integer if there exists an integer k such that r = 3k + 1. Prove that if a…

A: x is a type 1 integer. So, x = 3k+1, for some integer k. So, x² = (3k+1)²

Q: Prove that 3n^2 + 1 is even if and only if n is an odd integer.

A: We need to prove that 3n2+1 is even if and only if n is an odd integer.

Q: Prove: If Am = O for some positive integer m, then I − A is nonsingular.

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Q: Prove that if the sum of S = a1 + a2 + … + an is an even integer, and all the terms a1, a2, …, are…

A: Given S=a1+a2+…+an is an even integer and all the terms a1, a2,…,an are odd integers. To prove that…

Q: there are no integers x and y such that x is a prime greater than 5 and x=6y+3

A: Given query is to check the statement whether it's possible or not.

Q: Prove that for every positive integer n the number (1 – /2)" is not ratioanal.

A: Given: The given number is, (1-2)n where n is the positive integer.   To prove: The number (1-2)n is…

Q: Prove by any means that n!/(n − r)!r! is always an integer for 0 ≤ r ≤ n

A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

Q: Prove that 74n+3 + 2 is a multiple of 5 for all non-negative integers n.

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Q: Given positive integers k and m, there exists an integer N ≥ m such that

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Q: Prove that there exists no rational number x such that x2 = p, where p is a prime number (A prime…

A: We need to prove that there exists no rational number x such that , x2 = p  ,  where p is prime…

Q: Prove that if a and b are real numbers with a < b, then there are infinitely many rational numbers…

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Q: 18. a. Prove that for every integer a, if a' is even then a is even. b. Prove that V2 is irrational.

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Q: Prove that n^2 is an even number if and only if 3n + 1 is an odd number by proving  If n^2 is even…

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Q: Let n be a positive integer. Prove that for all x € Z, if x" is odd, then x is odd.

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Q: Prove that for any positive integer n, Vn is either an integer or irrational.

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

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