1. Let a and b be natural numbers, where a, b> 2, such that god(a, b) I and ab = (x + y)" for some natural number n and integers x and y. Prove that x+ y cannot be prime.

Q: Let i be a positive integer and A, [-i+ 1,1]. Then U14, %3D O R O None of these O F1,11

A: Given that 'i ' be a positive integer and Ai=-i+1,1. We have to find the union ∪i=1∞ Ai.

Q: Let a,b be integers, not both 0. Show { na + mb : n,m ​​​​​are integers} = iff a and b are…

A: We have to prove the given statement.

Q: Let "xy" denote "x does not divide y." Prove by any method. Let a and b be integers. If 3 ab, then…

A:

Q: [1+6n A" 7 4 4n -9 -5 -9n 1-6n | If , show that for every n natural number.

A:

Q: Given a positive integer n, let a2,n–1 a2,n a3,n-2 a3,n-1 a3,n An = аn-1,2 аn-1,n-2 аn-1,п-1 @n-1,п…

A: Given: Here the given positive integer n, Let An = 00...00a1,n00...0a2,…

Q: what

A:

Q: a) Find integers a and b such that a + b =a-b (mod 5).

A:

Q: a) Prove that every integer can be written a sum of powers of 2. b) Write 14, 17, 27 as a sum of…

A: We have to prove the every integer can be written a sum of power of 2 and write 14, 17, 27 as a sum…

Q: Let c, a, B be nonzero real numbers such that c + B. Find a given that 122n-3 2 cn-4 ac? n-D3 с — В…

A:

Q: 7. Ug = kuz-1¬- Ug–2, for every integer k > 3 uj = 1, u2 = 1

A:

Q: Show that there is exactly one positive number u such that In(u² + 1) + Tu = 2021.

A:

Q: 4) If a is an odd integer, then a? = ?? (mod 8) a) 0 b) 1 c) 2 d) 3

A: We have to find a such that a2 = ??(mod 8)

Q: Write down the supremum of the set {7+ 7/n2 − 5/m | n,m ∈ N}. prove ur answer from the definition.

A:

Q: 1) Prove that the numbers -197, -239 are prime numbers 2) Find all prime numbers between -90 and 90.

A: number system

Q: alb then albc for all integers c.

A: Since, a|b, a divides b or there exists an integer k such that b=ak.

Q: Let i be a positive integer and A, = [-i+1,1]. Then U, A= O [-1,1] O None of these O l-00,1[ O…

A: Given that          for i be a positive integer…

Q: III) Are there positive integers x and y suct that x³ = 2² +15? (Hint: for every integer x, x³ 0, 1,…

A: We shall show that no such positive integers exist.

Q: If a and b are positive integers and p is prime with p ł a, then the solutions of the linear…

A: We will check whether the given statement is true or false.

Q: If A = 0 and Ak = 0, for some posit ive integer k , then A must be singular Select one: O True O…

A: The solution is :

Q: let a, b, and m be integer, prove that if 2a+3b>=12m+1, than a>=3m+1, or b>=2m+1

A: To prove: For integer a, b, and m if   2a+3b≥12m+1Than, either a≥3m+1, or b≥2m+1.

Q: Let c, a, ß be nonzero real numbers such that c + B. Find a given that 00 162n-3 ac² cn 4

A:

Q: Prove that if sum of two numbers is irrational then at least one of them is irrational. Use indirect…

A:

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 a, b, c, d, e, and f be positive integers. Let d = gcd(a, b, c), e = gcd(a, b) and f = gcd(e,…

A: Given that for positive integers a,b,c,d,e,f , d=gcda,b,ce=gcda,bf=gcde,c

Q: p, q, pq | (apq − ap − aq  + a) for all a.

A:

Q: Find UA, and A, if for every positive integer i, i=1 1) A;= {-i, -i+1,...,-1,0,1,2,...,-1,i} 2) A =…

A:

Q: Let a and b be two positive real numbers. Let p > 1 and q> 1 be such that + = 1 (q is called the…

A:

Q: 16. Prove that an integer (a,-1an-2 · . . ao)10 is divisible by 11 if and only if ao + a2 + a4 + . .…

A:

Q: Let а, b and c be integers. Use the method of proof by contradiction to prove that if a|b and a f c,…

A: We have to solve given problem:

Q: prove the following statement. let a, b, and c bre a pythagorean triple of integers, i.e…

A:

Q: Let Z be the set of all integers and Q be the set of all rational numbers and R be the set of all…

A:

Q: If 3x = 6 mod 9, when -2 <x< 8. Then O x=(-1, 3, 5} x=(-1, 2, 5} O x=(-1, 2, 3} O x=(-1, 5, 6}

A:

Q: Let n and m be integers. Prove that if m + n >= 100, then either m >= 50, or n >= 50.

A: Prove the statement by proving the contrapositive. The contrapositive of the given statement is as…

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

A:

Q: Negate the statement: For all positive integers, n, there exists a positive integer, k, so that…

A: The negation of a statement is:- Statement:- p implies q. Negation:- p does not imply q.

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: I'm supposed to evaluate the proof, edit if there are errors or confirm that it is correct

A: We are given a proof for the given proposition The proof is correct, No changes/corrections  are…