Equivalence classes of mod 6 on Z have: a)  3 elements each b)  infinitely many elements each c)  6 elements each d)  1 element each

Q: Exercise 10. Show that if AC B, then ACB.

A:

Q: Given 6x + 2 ≡ 101 (mod 7), find an integer z such that 2x ≡ z (mod 7)

A:

Q: 25 Perform the mndi.cated -Computation Write the answer in scienh fic notation (4~10*/2.1

A:

Q: 3. Prove that for each natural number n, 十2 ...

A:

Q: Let n>,2 and k be any boretine Proce that integers (n-1)* | n, 2

A: Given, n≥2 and k is any integer. n≡1modn-1Therefore, ni≡1modn-1 ∀i∈ℕ…

Q: Establish by mathematical induction that any postage of at least 12c can be obtained using 3c and 7c…

A: We have to establish by mathematical induction that any postage of at least 12c can be obtainedusing…

Q: If positive integers are chosen at random, what is the minimum number you must have in order to…

A:

Q: 4. Find two distinct fractions which kiss 5. The natural representatives mod 11 are The minimal…

A:

Q: Construct a proof that ZEIF = ZGIH. н

A:

Q: If z is an odd ratural number then 2^+1 is divisi ble by 3 hìnt: prove by Induction

A:

Q: theorem Every Ving Can be imbeobling în aving with

A:

Q: Any two integers are congruent modulo .when they are both even or both odd. Least common multiple…

A: Answer : smallest prime n=2

Q: Prove that a^561 ≡ a mod 561.

A:

Q: Any two integers are congruent modulo .. when they are both even or both odd. Least common multiple…

A: Given a statement

Q: Recall that tn denotes the nth triangular number. Prove the following identity: tn+2k = tn (mod k)

A: If tn is the nth triangular number, then tn=nn+12

Q: Prove that multiplication is associative in Q.

A: Prove that multiplication is associative in Q. Let r, s, t be in Q. Then  r=ad, s=be, t=cf; such…

Q: Show

A:

Q: The linear congruence ax = b(mod m) has a unique solutionif and only if: gcd (a,m) =1 gcd(a,m) =1 o…

A: First option is correct. gcd(a, m)=1.

Q: Using induction, prove that for all n > 1,

A: To prove :              For all n > 1,              A1 + A2 + ... + AnT = A1T + A2T + ... +…

Q: 4 Prove that סלא is Riomonn inkgpable

A:

Q: what are the congruence classes in the set of intergers?

A:

Q: b. Prove that the integer 359,240,758 cannot be a perfect quartic. (i.e. fourth power) The 758 ppose

A:

Q: Give three elements of the equivalence class of 15 modulo 7

A:

Q: 6. Prove by CVI that every natural number n > 2 is a product of prime numbers. NOTE. A prime number…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Prove that the Ving of ComPelX numbers C¢,+.)is fielol

A: Solution is given as below:

Q: Determine what number the product is congruent to in mod 3. 5.8 5.8 = (mod 3 )

A:

Q: 3. Prove that 2" > n For all nuatural numbers

A: Solution

Q: Let n, m, a, b be positive integers. motl Lof oontredietion u th

A: Given, n, m, a, b be positive integers. To prove that…

Q: Use Fermat's theorem, show that if gcd(a, 155) = 1 then a121 = a (mod 155). %3D

A:

Q: Prove that the nth Fibonacci number fn≤αn−1, where α is the golden ratio.

A:

Q: 2. The binary operation * defined on Z by x * y = 1 + x + y on the set of real numbers. Show that *…

A:

Q: Show that the operation a*b=1+ab on the set of integers Z, has no identity element

A: Let e be the identity element which exist.  Then, a*e = e*a = a

Q: Prove the fallowing egudion for every n,2 by hathematical enduelion. 35.6+35 6+ 35.63+ -+35.6"= 7.6"

A:

Q: (b) Prove that multiplication mod n is commutative.

A:

Q: b) Prove that if n2 (mod 3) = 1 then n is not divisible by 3.

A: The required proof is as follows -

Q: 2. Prove that võ is iational number. 3. Ifa,beR. prove that ab

A: ANSWER 2:

Q: Show that the set of a+bi Gaussian integers has the same strength as the set of natural numbers N.

A:

Q: Prove by CVI that every natural number n > 2 is a product of prime numbers. NOTE. A prime number p…

A: Every integer greater than 1 can be written as the product of prime numbers. In other words, every…

Q: If g is a primitive root modulo 23, then for which 1 si< 23 is g' also a ive root?

A: Answer:

Q: Show that af +...+ a, = a1 + ...+ ap(mod p).

A:

Q: Prove in any Pythagorean triple (a, b, c) that either a or b is divisible by 3 and one of a, b, c is…

A: A Pythagorean triple is three numbers a,b,c such that a2+b2=c2. Also if a,b,c is a Pythagorean…

Q: 4. For natural numbers n and k, prove (C) - ()

A: We have to prove

Q: Exercise 5. Prove that given positive integers a and b, there is a positive integer d such that • d…

A: Given two positive integers a and b, let us choose d = gcd(a, b) [greatest common divisor] Then…

Q: 5. (a) Use modular arithmetic to show that if an integer a is not divisible by 3, then a2 = 1 (mod…

A: To prove the given statements (using arithmetic modulo 3)

Q: Prove that tim 51

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

Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…

Q: If p,q,r are prime number but p is not equal to q, then what is gcd(p,q)? If p|rq, prove that p|r…

A: Given that p, q and r are prime numbers and p≠q.