If the sum of any two integers is even, then so is their difference.

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

A: x is a positive integer. Find sum of 6x and 11x is divisible by what?

Q: How do I prove that if a, b, and c are integers with c not equal to 0 then ac = bc => a = b?

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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 by contradiction that for any positive two real numbers, x and y, if x·y ≥ 100 then either x…

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Q: pair of these integers with a sum equal to 9.

A: Let us take set under consideration {1,2,3,4,5,6,7,8}Possible pairs {{1,8}, {2,7}, {3,6}, {4,5}}…

Q: Suppose that nine times a number is added to 63 and the result is positive. What are the possible…

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Q: A natural number greater than 1 that has only itself and 1 as factors is called a/an number.

A: Given that  A natural number greater than 1 that has only itself and 1 as factors is called…

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

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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…

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Q: of the first 150 positive integers?

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Q: Find a pair of numbers that provides a counterexample to show that the given statement is false.…

A: We know that, The product of two even counting numbers is an even counting number. The product of an…

Q: For all integers x and y, if 6x+y is odd, then x and y are both odd. True False

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Q: Is there a real number that is bigger than all integers? If so what is the number and if not what…

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Q: Suppose that ƒ(0) = 5 and that ƒ'(x) = 2 for all x. Must ƒ(x) = 2x + 5 for all x? Give reasons for…

A: given: f(0)=5 f'(x)=2

Q: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

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Q: Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

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Q: If a + 0, the degree of ax" is

A: The degree of a polynomial is given by the highest exponent appearing on the variable term. For…

Q: Prove that it is not true that the product of two irrational numbers is irrational.

A: To check:   Whether the product of two irrational numbers is irrational or not.

Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

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Q: Every odd integer is the sum of three odd integers. There exist an irrational number c and a…

A: True  proof Let us assume  that x is even So x=2z for some constant z Therefore x.y=2zy Which…

Q: There is a positive integer whose square is equal to itself. real а. There is a number m such that…

A: To rewrite the statement in a different way: Firstly, rewrite the statement in the given statement…

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: "Every positive integer can be written as the sum of the square of 3 positive integers." prove…

A: Given: "Every positive integer can be written as the sum of the square of 3 positive integers." To…

Q: Prove that the product of two odd numbers is odd.

A: We first define a general odd number. Lets say that n is an integer. When we multiply any number by…

Q: Prove by contraposition If a sum of two real numbers is less than 50, then at least one of the…

A: To prove the required statement by contradiction

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it…

A: (a)The fill up for the statement to rewrite the given statement: Is there an integer n such that n…

Q: Two integers, a and b, have a product of 48. What is the least possible sum of a and b? The least…

A: Answer is a+b= -49. Explanation is given below...

Q: If the sum of two counting numbers is an even number, then the product of the two counting numbers…

A: ANSWER:- 1) From the given statement, If the sum of two counting numbers is an even counting number,…

Q: Can you please explain to me why is (k(k+1) ) / 2 is an integer and I know it always even so can you…

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Q: If a and b are nonzero integers and m is the least common multiple of a and b, prove that (a) ∩ (b)…

A: Consider the given statement. (a) ∩ (b) = (m) Consider the left side of the equation. As, m is the…

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…

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Q: Prove that the sum of the squares of three consecutive integers cannotbe a square.

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Q: Prove that it is impossible to choose three distinct integers (a – b)|(b – c), (b – c)|(c – a), (c –…

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Q: Suppose that J and K are points on the number line. If JK = 18 and lies at - 3 , where could K be…

A: NOTE: Refresh your page if you can't see any equations. . the distance between J and K is 18 and the…

Q: If n is an even integer, then 3n° is an even integer.

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Q: The sum of an integer and 4 less than that integer is even.

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Q: Assuming every integer is either even or odd, How would I show that for any integer, the difference…

A: Assume that every integer is either even or odd. Let the integer considered say x is even. Then by…

Q: How large a subset of the integers from 1 to 50 must be chosen to guarantee that one of the numbers…

A: We have to determine How large a subset of integers from 1 to 50 must be chosen to guarantee that…

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: Find five consecutive odd integers such that when the fifth is subtracted from eigt times the…

A: We're going to solve

Q: Let x, Y be positive real numbers. If x2 – y² =1, then x or y (or both) is not an integer.

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Q: Find a pair of numbers that provides a counterexample to show that the given statement is false.…

A: Given statement: If the sum of two counting numbers is an even counting number, then the product of…

Q: Prove by contradiction that if a sum of three real numbers is less than 30, th at least one of the…

A: We have to prove by contradiction that if a sum of three real number is less than 30, then at least…

Q: Suppose x is a real number and x is not equal to zero. Prove that if ((x^1/3)+5)/((x^2)+6)=1/x, then…

A: Given that, x is a nonzero real number, and 8 is not a root of given equation.

Q: the sum of any positive integers that is less than 30

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Q: The sum of the four positive integers a, b, c, and d is at most 10. Find the number of possible…

A: Explanation of the answer is as follows