prove the difference of the squares of two z integers is always divisible by 16 (i.e. z1^2 - z2^2 = 16) z = 4k+2 for some integer k

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: 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: 5. Show that Z[V-6] is not a UFD by factoring 10 in two ways. Why does this prove that Z[V-6] is not…

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Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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Q: Suppose N = Fox F₁x F2, where the three factors are pairwise prime; hence, each factor and the…

A:

Q: 16. All positive integers with negative squares are prime.

A: It is a statement. And it is  vacuously true.

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

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Q: 4. Describe all zero-divisors and units of Z6 O ZOQ

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Q: Use a direct proof to show that every odd integer is a difference of two squares. (For example: 1 =…

A: first we take a odd integer then we write in 2y+1 form then we transfer to difference of two…

Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: The sum of any two integers of the from 4k + 1is always: prime Zero Od Even

A: Given statement, The sum of any two integers of the form 4k + 1 is always……

Q: Which term of the expansion of (x° + 7) contains x 28 ,36 Term number contains æ36

A:

Q: Suppose, and no are the smallest of two positive integers so that 300x is a perfect square and 300y…

A: The given problem is to find the values of x and y and sum of value of x+y for which smallest…

Q: Prove that 2^10 + 5^12 is a composite number.

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

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Q: Can you show me step by step how to do a proof by contradiction for the theorem that if the result…

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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 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 3√3  is irrational

A: See the attachment.

Q: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

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: 2. Prove that the Cartesian product N Nis countable

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Q: 3. How many zeros at the end does the number 2022! have? (n! is a product of all positive integers…

A: Given,Number = 2022!To find: Number of zeros at the end (trailing zeros) of the given number  As per…

Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: How many integer solutions to x1 + x2 + x3 + x4 = 32 are there for which x1≥6, x2 ≥5, x3 ≥3 and x4…

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Q: Number theory

A: Consider the provided question,

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: SHOW THAT 3^2N - 1 IS DIVISIBLE BY 8 FOR ALL POSITIVE INTEGERS.

A: We prove this by induction.

Q: Prove that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

Q: the 20 1, 2, 3 Factor the number 211 – 1 by Fermat's factorization method. -

A: Fermat’s Factorization method is based on the representation of an odd integer as the difference of…

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: Prove that 4 does not divide n2 +2 for any integer n.

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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: Show that 2^n >n^2 if n is large enough.

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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 3/3 is irrational.

A: A number which can be written in the ratio , where and are integers are called rational numbers.…

Q: Prove that 11 divides 3* 5k + 42k+1 + 33k+2 – 5 * 72k for every k non-negative integer

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Prove that 3+√3 is irrational.

A: As per guidelines we are supposed to solve the maximum one questions at  a time kindly post another…

Q: Prove that no number of form 4k+3 (where K is a positive integer) can ever be the sum of two perfect…

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Q: Which term of the expansion of (x + 7)° contains x° Term number contains x°.

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

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Q: Prove that 3" + 1 is divisible by 4 whenever n is an odd positive integer.

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Q: Obtain the expansion of : 2 up to the term in x8 1-3x2

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Q: Vn € Zt [obeb(n, n + 1) = 1]

A: l  let n,n+1 are two consecutive positive integers let  the greatest common divisor i.e…

Q: 9. Prove that the greatest common divisor of two positive integers divides their least common…

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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…