Show that 2 is a primitive root of 19.

Q: Show that the square roots of two successive natural numbers greater than N2 differ by less than…

A:

Q: Find the roots Lzerel of eah tunction and State tHhe multipicty of any veputed root - 니 2x

A:

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: Find all possible digits that can be placed in the space so that the numbers is 1,182,1_4

A: Apply the rule for a number divisible by 9.

Q: 10. Show that the product of a+bi and b+ai always results in a purely imaginary number, given that a…

A: Given two complex numbers: a+bi and b+ai

Q: How many real sixth roots does 64 have?

A: We have to find real sixth roots of 64.                      x^6=64   We have to find the real…

Q: 58° of 2.

A: Given: A figure with one of the angles as 58°.

Q: Show that √2 + √3 is irrational.

A:

Q: Is the statement true or false? If an odd number is less than 15, then it is prime. False Give a…

A: We have to check

Q: 14. Demonstrate that 21 has no primitive root.

A: An integer n≥2 has a primitive root if and only if the the number n is one of the following; 2,4,…

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: 1. Prove that the square root of 2 is not a rational number.

A: Definition of rational number : If x is a rational number then it can be written in the form of pq…

Q: 2X=3 mod 5 Show your work

A: the given congruence equation is 2x≡3mod 5 the general congruence equation is ax≡bmod c…

Q: Prove that Q(3^(1/3), sqrt(3))= Q(3^(1/6)), where Q is the set of all rational numbers.

A: First observe that,

Q: Check the definiteness of the following quadratic form;

A:

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

A:

Q: Find the primitive roots (mod 17).

A:

Q: Prove by contradiction that 8√2/ 7 is irrational. ( 8 root 2 by 7)

A: To prove  827  is irrational, We have to show that 2 is irrational, As we know that when multiple…

Q: Characterize the primes p for which 63 is a quadratic residue.

A: Quadractic residue: Let p  be any odd prime number and gcd(a,p)=1. Then a is said to be a quadratic…

Q: 5 is a quadratic non- residue mod 17 True False

A: 5 is a quadratic non- residue mod 17

Q: Let æ be a real number. Show that [3r] = [a] + æ + + 3

A: Here we have to show that 3x=x+x+13+x+23

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: Are the irrational numbers closed under multiplication?

A: Use closure property.

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: Va 占>0 !! に

A:

Q: If k is a positive integer, prove the k root of 6 is irrational

A: Given:Given that k is apositive integer.To find:The kth root of 6 is Irrational.

Q: Prove by contradiction that, if n is a positive integer which is not a perfect square, the sqrt(n)…

A:

Q: Use proof by contradiction to show that there is no positive real number x such that (x-2)/x ≥…

A: In an inequality, if a positive number is multiplied then there is no change of sign of inequality.…

Q: Suppose that m andn are positive integers and b is a real number so that the principal n" root of b…

A: We have been given that nth root of b exists where b is the real number.

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: For which positive integers a is the congru

A: Given,               ax4≡2 (mod 13)

Q: Calculate all the primitive roots of 41 and 26.

A: (a) Consider a=41. Therefore: φa=φ41=411-410=40 Now φa=40 can be written as: φa=40=23×5. Therefore…

Q: Show that for every real number ɛ 8 > 3ɛ. greater than zero, there exist a real number 8 such that

A:

Q: Say whether the Runction is even, odd, or neither. Give reusans for your dnswer. go0 = x"+x-1

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: When a is a positive real number, the ________ ________ root of −a is defined as √−a = √ai.

A:

Q: Complete the steps by app that the left hand side of th the right hand side. Nec in a ot

A: To prove.

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

A:

Q: The set of all positive multiples of v2 is countable.

A:

Q: Prove the given statement: The sum of the squares of two odd integers cannot be a perfect square.

A:

Q: Prove that the simple continued fraction of a real number is unique i.e. if [a0,a1,...] = x =…

A: To Prove: The simple continued fraction representation of a real number is unique, i.e. if…

Q: Express the following using factorials: 10 x 9 x 8 x 7

A: Meaning of factorial of a natural number : Suppose n is a natural number then meaning of n! is…

Q: Let x be a real number. Prove by contrapositive that if æ + 7x = 90, then x < 4.

A:

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: #1. Find all five 5" root of Z = 32cis40° 5th

A: Given z= 32 cis 40° rewrite it as z= 32cos 40+isin 40 if z= rcos θ+isin θ then n distinct complex…

Q: Prove that if æ is a positive real number, then

A: To prove that if x is a positive real number, then                               x=x

Q: How many multiplicity of - 1 are there in the roots of x +3x2 + 3x+1 0? O 1 O 2 3 O 4

A: Given equation  x3+3x2+3x+1=0

Q: There are no primitive roots modulo n = 128. O False True

A: Theorem: There are no primitive roots modulo 2n for n&gt;2.

Q: Prove that cube root of 2 is irrational by contradiction

A:

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

A:

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.