Suppose x and y are any integers. Prove that if x and y are odd, then x + 5y? is odd.

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

A:

Q: Prove thq+ for all integer a, b, and c if 9+b of a and k is even.

A: Introduction: Even and odd numbers are also referred to as: Even Number: An even number is one that…

Q: Prove that for any integer a, 3 divides

A: To prove that for any integer a, 3 divides (a-3)(a+4)(a+2).

Q: x and y. Prove that if alb and alc, then a|(xb+ yc) for any integers

A:

Q: Determine (with proof) if there exist integers m and x such that 2003m 2 = 2.x(x – 20). If they…

A:

Q: Find all integers z, positive or negatives, such that z2 + z is prime.

A:

Q: Let a, b be two positive integers with (a, b) = 1. If ab = c, then prove that there exists

A:

Q: If all entries of A are integers, and det A = 1 or -1, prove that all entries of A -1 are integers.…

A:

Q: Prove that there are no positive integers a and b such that a + b³ = 250.

A:

Q: If x is an odd integer, then x² +3x+5 is odd.

A:

Q: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

Q: Suppose there is a real number r with the property that r + 1/r is anodd integer. Prove that r is…

A:

Q: Prove that if x is a real number, then ⌊2x⌋ = ⌊x⌋ + ⌊x + 1/2 ⌋.

A:

Q: Prove that the following statement is true for all positive integers m and n:  m and n are…

A: m and n are multiples of each other if and only if m = n. for all positive integers m and n:

Q: Prove that 2 + v3 is irrational.

A: Given: To prove: 2+3 is irrational.

Q: Let a, b eN. Prove that if a + b is even, then there exists nonnegative integers x and y such hat x²…

A:

Q: Prove that for every positive integer m, there is a positive integer k such that k does not divide m…

A: Given:-                 prove that every positive integer m, there is a positive integer K such that…

Q: Prove or disprove the following theories by using proof by constraposition method. Note that the…

A:

Q: Prove by CONTRAPOSITIVE if x is an odd integer, then x²+3x+5 is odd

A: Explanation of the answer is as follows

Q: Prove that if æ is a real number, then [2a] = [x] +

A: If x is a real number then  to prove- 2x=x+x+12 Greatest integer function is denoted by [x] for any…

Q: 3. Prove by contradiction, that if x and y are odd integers, then there is no integers z such that…

A:

Q: Use proof by contradiction to show that no integers  y and z exist for which 24y + 12z = 1.

A:

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

A:

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: Use the contradiction method to prove: There exist no integers a and b for which 21a + 30b = 1

A:

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: If α is even, prove that α-1 is even. If α is odd, prove that α-1 is odd.

A: We know that the multiplication of 2-cycles can help us find any permutation of a finite set, so we…

Q: Determine (with proof) if there exist integers m and x such that 1997m + 47x2 = 3(17x – 1). | If…

A: Given equation is 1997m+47x2=317x-1. To determine: Whether integers m and x exists or not.

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: Suppose that k and n > k are fixed positive integers. Justify the identity k+1,

A: Given k and n≥k are fixed positive integers.

Q: x]+n x+n m m

A: By using definition of greatest integer function, For x∈ℝ we have, x=x+x, 0≤x<1 where, x is…

Q: Let x and y be integers. Prove by the contrapositive that if x2 - y2 is even, the x + y is even

A: Let x and y be integers such that x2-y2 is even. We have to prove that x+y is even. Suppose that x+y…

Q: Prove that if a and b are both odd integers, then 8 (2a2+36²).

A:

Q: Consider the statement: "there are no integers and such that 20a+4b%3D1." Is this true or false?…

A:

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

A:

Q: Use a direct proof technique to prove the following theorems: For all integers z and y, r² + y - 3…

A:

Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

A:

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: Let x be an integer. Prove or provide a counterexample: If 4x+7 is odd, then x is odd.

A: We provide a direct proof and give counter example to show that given statement is false.

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 the following by contrapositive: If x is an even integer, then x^2 is even.

A:

Q: Let x be positive integer. Prove If x2 + 1 is even, then x3 is odd

A: Consider the provided question, Let x be positive integer. We need to Prove that, If x2+1 is even,…

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

A:

Q: Suppose a, b, and c are integers and suppose a b and ałc, where atc denotes that c is not divisible…

A:

Q: Select each step to complete a proof by contrapositive of the theorem below. Theorem: For every pair…

A: Here we have to prove the theorem that “For every pair of integers x and y, if 5xy + 8 is even, then…

Q: Prove that ifx^2−6x+ 3is odd, then x is even

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.