2 There are integers m and n such that m² +n² is a perfect square.

Q: b) Use an English proof to prove that if integers m and n are perfect squares, then m*n must also be…

A: Consider the provided question, Given, m and n are both perfect square. We need to prove m*n is also…

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: Discrete

A:

Q: Prove that 2 is a factor of n2 + 5n for all positive integers n.

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: 1) Show that the square of an odd integer is odd. 2) Show that the product of two odd integers is…

A:

Q: Prove that τ(n) is odd if and only if n is a perfect square.

A: From the definition of τ(n), it represents the number of positive divisor of n. Let τ(n) be odd…

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: Prove that if n is odd, then 2 is not a square in GF(5n). In other words, prove that there is no…

A:

Q: 4. Show that if k is a triangular number, then 8k + 1 is a perfect square.

A: Since you have posted a multiple question according to guildlines I will solve first question for…

Q: what is (a'+b,a+b) Wher e a and b are relatively prime integers that are not both Zero1

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: Prove that there do not exist three consecutive integer values of nn for which 43n+1443n+14 is a…

A: Given 43n+14 We have to Prove that  ∄  three consecutive integer values of n for which 43n+14 is a…

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

A:

Q: Prove that the sum of the first n cubes is equal to the square of the sum of the first n positive…

A: According to the given information, it is required to prove the following.

Q: by p. 13, and 17. 2 For n= 1, 2,..., 10, show that 5u +4(-1)" is always a perfect square. 3. Prove…

A:

Q: Show that the cube for any integer is of the form 7k, or 7k ± 1

A:

Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

A:

Q: Which statement is not correct? Sum of two irrational numbers is irrational. O For every integer n,…

A: The incorrect statement is : sum of two irrational numbers is irrational.   the detailed solution is…

Q: Use the Division Algorithm to prove that (a) The square of any integer is either of the form 3k or…

A:

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

A:

Q: 20. (a) Fill in the blank: 1+3+5+..+ (2n – 1) = k=1 (b) Use part (a) to prove that the sum of the…

A: This is an arithmetic series with common difference (3-1)=2 And this has n terms in it So we use the…

Q: Prove that if n is odd, then 2 is not a square in GF(5n).

A: Given n is odd number We have to show that 2 is not a square in GF5n. We know that the field GF5 is…

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: Theorem 2.9. Let n be a positive integer and write it as n = N²m, where m is square-free. If m…

A: n be a postive Integer and wollte it as mn = N ^ 2 * m where m is square furee. fm Contains no…

Q: R.2. Let S represent the set of positive integers that are perfect squares, and let C represent the…

A:

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: Prove that if n is odd, then 2 is not a square in GF(5n) In other words, prove that there is no…

A: PROOF :

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

A:

Q: (b) Every integer n> I is the product of a square-free integer and a perfect square. [Hint: If n =…

A: By the fundamental theorem of arithmetic for an integer n>1 n=p1k1·p2k2.....prkr where each pi,…

Q: Prove that for all primes p2 5, p + 1 cannot be a perfect square. Recall that is a perfect square if…

A:

Q: b) Hence or otherwise prove that there do not exist three consecutive integer values of n for which…

A: First let us assume that 29n+5 is a perfect square for some consecutive integers n and n+1. Then…

Q: Let a and b be integers,not both zero .Then a and b are relatively prime if and only if

A: Solution are as follows

Q: Number theory.

A:

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

A:

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

A:

Q: 15.5 Prove that if n is a perfect square then n = 4q or n = 4q + 1 for some q e z. Deduce that…

A:

Q: 2. ) Show that the cube of any integer is of the form 7k or 7k +1 b) Verify that if an integer is…

A: 2) a. To show that ''The cube of any integer is of the form 7k or 7k±1''

Q: prove the difference of the squares of two z integers is always divisible by 16 (i.e. z1^2 - z2^2 =…

A: We prove that (z_1)^2-(z_2)^2=16k for some integer k, then we say that divisible by 16.

Q: 4. Prove that the cube of any integer can be written as the difference of Notice that squ n³ = (1³ +…

A: To prove that cube of any integer can be written as the difference of two sqaures.we have given hint…

Q: Show that if p is an odd prime, then there are exactly (p − 1)∕2 quadratic residues of p among the…

A:

Q: Consider the following universal statement. The sum of any two perfect squares between 1 and 20 is…

A: A number made by squaring a whole number is a perfect square. According to question, We have to find…

Q: Show directly that for p a prime integer, p can be written as p=(a)^2 - d(b)^2 then d is a perfect…

A: From the given statement, p be the prime factor and p can be written as p = a2 – db2.To show: d is a…

Q: Prove that 2 is a factor of n2 + 5n for every positive integer n.

A: To prove: 2 is a factor of n2+5(n) ∀n.

Q: Give a direct proof that if m and n are even are both perfect squares, then mn is also a perfect…

A: Given that, m and n are even and both perfect squares.

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

A:

Q: 1. Use Direct Proof to prove that the product of two consecutive integers is divisible by 2.

A: We have to prove that product of two consecutive integers is divisible by 2.

Q: 1. Show that if then p and q have no non-trivial common factor. and q are natural numbers and there…

A:

Q: Prove that the square of any odd integer n, (n =4q +3 for some integer q) has the form 8m + 1 for…

A: Here we have to prove that any odd integer n which is of the form (n = 4q + 3) , where q is an…

Q: 4. Prove that an element r of Z2 \ 2Z2 is a perfect square if and only if x = 1 (mod 8).

A: we prove first part⇒let x∈Z2\2Z2 is a perfect square whereZ2\2Z2…

Q: [2.1.22] I have 51 rectangular pieces of cardboard, each of which has an integer length and width in…

A: Given: Let's consider I have 51 rectangular pieces of cardboard in set 1,...,100, each with an…

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: