2. Let meN and a € Z. (a) If ged(a,m) = 1, then Bézout's lemma gives the existence of integers x and y such that ax + my = 1. Prove that a+mZ is the multiplicative inverse of a +mZ. (b) Determine the least nonnegative integer representative for (11+163Z)-¹ by expressing 1 as a linear combination of 11 and 163 (using the extended Euclidean algorithm).

Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: The greatest common divisor (gcd) of non-zero integers x and y is the the smallest positive integer…

A:

Q: greatest common divisor(g.c.d)

A:

Q: Let A = AB= 1 -1 2 0 5 1 -2 0 −1 14 -57 3 [-2 27 -32 0 -5 2 -1 1 0 -7 1 X -2 and B = 5 2 Find x such…

A:

Q: Find the number of ordered pairs (m,n) such that m and n are positive integers in the set {1, 2,…

A: Let us consider Tm=2m+1.  m Tm 1 3 2 5 3 9 4 17 5 33 6 65 7 129 8 257

Q: Prove that the product of two integers, one of the form 3k₁ +2 and the other of the form 3k2 +2,…

A:

Q: The decimal representation of a positive integer a is given by a = an-1ªn-2 *** α₁α where a =…

A:

Q: how the element 2+√3 has a multiplicative inverse in Q(√3). Show that Q(√3) is a subfield of C

A:

Q: Use the factorization A = PDP-1 to compute AK, where k represents an arbitrary positive integer. 35…

A: (.)  Given ,       -815-611=35232001-3   5  2-3 Where , A=-815-611   ,   P=3523  ,  D=2001  &…

Q: Exercise 4.6. Prove that if h1, h2 are two greatest common divisors of f1, ..., fs, then there…

A:

Q: Show that if a and b are coprime integers, then gcd(a + b, a² – ab + b²) = 1 or 3.

A:

Q: Find the greatest common divisor for 102 and 66, (102, 66), and the values of m and n such that…

A: To find the greatest common divisor of 102 & 66. For that we can use Euclidian Algorithm.…

Q: Let a, m, and n be positive integers with a>1. Prove that am – 1 | an – 1 if and only if m | n. For…

A:

Q: (9) Prove that for integers x, y and z, if x² + y² = z² then either x or y is even.

A: We use method of contradiction to prove given statement.

Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: Exercise 6.4. We say that a number î divides another number z if there exists an integer k such that…

A: We say that a number x divides another number z if there exists an integer k such that xk = z. Prove…

Q: 8. Let K be an integer that is greater than 1, a perfect square, and equal to Ei for some integer D.…

A: A perfect square is a positive integer whose square root is a positive integer. Example of perfect…

Q: Let m and n be two relatively prime positive naturals, and consider what naturals can be expressed…

A:

Q: Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n.

A:

Q: Use the factorization A = PDP' to compute A", where k represents an arbitrary positive integer. -2…

A:

Q: prove that for m ≥ 3 an integer, ordm(m − 1) = 2.

A:

Q: Find the general solution of the recursive relation an = 2an-1-12an-2 for every integer n > 1. None…

A:

Q: 8. give their greatest common divisor. Let F = Z3. Factor the polynomials x³ + x² + 2x + 2 and x³ +…

A: Given the polynomials f(x) = x3+x2+2x+2 and g(x) = x3+1.