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

Q: Prove that the difference of squares of any two odd numbers is multiple of 8. (SOlve this task…

A: To proveDifference of squares of any two odd numbers is multiple of 8.

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: Prove the square of any integer has the form 4k or 4k+1 for some integer k

A: Consider the set of all integers , To prove that the square of any integer is of the form 4k or 4k+1…

Q: Show that if a positive integer is odd, then it's square is odd.

A: To show that if a positive integer is odd then its square is odd

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: 2. Prove that the product of any two odd integers is an odd integer.

A: Let x and y be two odd numbers. If a and b are odd, then a and b can be written as: x=2m+1 , y=2n+1…

Q: Prove that there does does not exist a unique positive real number, x, with the property that x…

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Q: Prove the square of any odd integer is expressible in the form 8n+1 with n ∈Z.

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Q: Prove that there is no rational number whose square is 12.

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Q: Show that every positive integer can be written as the product of a square (possibly 1) and a…

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Q: Find two positive integers such that the sum of the first number and five times the second number is…

A: Solution

Q: - Find two positive even consecutive integers such that the square of the smaller integer is 10 more…

A: Consider the first even positive integer be 'x'. The second consecutive even positive integer be…

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: Prove that there is no triangular number which is one less than a multiple of 11.

A: Introduction: A triangular number is a positive integer that is the average of two integers. For…

Q: 11. Prove that if n is a perfect square, then n + 2 is not a perfect square.

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Q: Find a counterexample to the statement that every positive integer can be written as the sum of the…

A: Claim: Every positive integer can be written as the sum of the squares of three integers. The given…

Q: Prove that there are 1000 consecutive positive integers that are not perfect squares.

A: Here I give an example. That proves the hypothesis.

Q: Prove that 67n+ 17 cannot be a perfect square for any three consecutive integers n.

A: We will use quadratic residue and proof by contradiction method.

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: Use direct proof to show that product of two consecutive integers is even.

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

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

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: The product two odd integers is even, whereas the product of an odd integer and an even integer is…

A: The product of two odd integers is odd. The product of an odd integer and an even integer is even.

Q: is an even integer then n square is .A Negative

A: Suppose n=m is an even integer.Since n=m , then n2=m2

Q: Prove that the product of an even number and an odd number is even.

A: Here, objective is to prove that the product of an even number & an odd number is even

Q: Prove that the difference of squares of two odd integer multiples of 3 is divisible by 72.

A: Odd integer: Any integer which is not divisible by 2 is known as odd integer.Ex: 1,3,-1,-3 and so…

Q: Prove that if n is a perfect square, then n +2 is not a perfect square.

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Q: Prove that the difference between the square of any odd integer and the integer itself is always an…

A: An odd integer is of the form 2n+1; n ∊ ℤ. Consider the difference between the square of an odd…

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: Find an integer that leaves a remainder of 2 whan divided by either 3 or 5, but that is divisible by…

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Q: Prove: The product of 2 numbers is equal to the product of their least common multiple and their…

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Q: if two even integers are multiflied, then the product is also even

A: Ans. True

Q: The sum of the squares of two odd integers is 1) Perfect 2) Prime 3) Not perfect

A: We will find out the required solution.

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: What two positive real numbers whose product is 11 & have the smallest possible​ sum?

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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: Prove that there does not exist an irrational number whose square root is rational.

A: Since, (p/q)=p^2/q^2 is rational. The square of any rational is rational, hence no rational is the…

Q: There is no integer that has a reciprocal that is an integer.

A: As per the instructions, we will be answering only the first question. For (32), Notice that the…

Q: Prove (NOT BY EXAMPLE) that for every nonzero integer, its square is positive.

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Q: Show that every odd integer is the sum of two consecutive integers

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Q: Prove that there exists an irrational number whose square is rational.

A: We have to show that there exists an irrational number whose square is rational

Q: Show that every odd integer is the difference between two consecutive perfect squares.

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Q: Prove that 13,050.046 is not the square of a natural number.

A: Use the fact that an even number is perfect square the number is divisible by 4

Q: "There is no largest integer."

A: Given: To explain the given statement as follows,

Q: How many integer solutions are there to 1 +X2+X3+X4+Xs=31 with

A: Given, x1+x2+x3+x4+x5=31 Number of Solutions when xi≥0 and xi>0.

Q: the student noticed that if he writes a multiplication sign between the second and third digit of…

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Q: Prove that there is not rational number whose square is prime

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Q: Prove that the difference of squares of any two odd numbers is múltiple of 8. (SOlve this task…

A: To prove- The difference of square of any two odd numbers is a multiple of 8.   Odd numbers are…