Suppose that m and n are relatively prime and r is any integer. Showthat there are integers x and y such that mx + ny = r.

Q: Prove that for all integers m and n,if m and m + n are even,then n is even

A:

Q: Prove the following statement. 1) if k is a positive integer, then 3k+2 and k+1 are relatively…

A: We have to prove: If k is a positive integer, then 3k+2 and k+1 are relatively prime.

Q: Use direct proof to show, "If m + n and n + p are even integers (s and t), where m, n, and pare…

A: To answer the given questions we shall use the following definition  An integer n is said to be even…

Q: 11. Prove that for each integer a, if a² – 1 is even, then 4 divides a² – 1.

A:

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: Use a proof by contradiction or by cases to show: No integers y and z exist for which 2y2 + z2 = 14

A: Result: Product of two positive term is always positive and product of two negative term is always…

Q: Prove that there exixts integers m and n such that 15 m + 12n = 3

A: Given equation is 15m+12n=3

Q: Show that if p is an odd prime, then every divisor of the Mersenne number 2p − 1 is of the form 2kp…

A:

Q: Prove that for all integers a and m, if a and m are the lengths of the sides of a right triangle and…

A: Given: a and m are length of sides of right triangle and m+1 is hypotenuse then a is an odd integer…

Q: show that for eury A s integer nz3; A n- Am-2 +A2J. Mence determine ASO.

A: To show: That An=An-2+A2-I for n≥3 then find A50. Given: The matrix is A=100101010. Calculation:…

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169.

A: Topic - principle of mathematical induction

Q: Prove that the following statement is true for all positive integers m and n:  m and n are…

A: m and n are multiples of each other if and only if m = n. for all positive integers m and n:

Q: Suppose x and y are any integers. Prove that if x and y are odd, then x + 5y is even

A: Suppose x and y are odd integers. To show x+5y2 is even. Note: Square of an odd number is odd.…

Q: Suppose that the prime factorizations of two positive integers x and y are expressed using a common…

A: The prime factorization is breaking down a number into the product of only prime numbers. These…

Q: Suppose a is an integer. Prove that if a^3 is a multiple of 3, then a is a multiple of 3.

A: Let a3 is a multiple of 3 but a is not a multiple of 3.a3 is a multiple of 3Therfore, 3 divides…

Q: For any distinct primes p, q,r and s, we have pq # rs. For any positive integer n, there exists n…

A:

Q: Suppose p is prime and a is a non-zero element of Z,. Prove that xP – x + a has no zero in Z,

A:

Q: ivided by 6.

A: Given, m is odd. So, m = 2k+1 for some integer k  &, n leaves remainder 2 when divided by 3  So,…

Q: contradiction

A: Please refer to the image below

Q: Suppose if p and q are odd integers with p!= q (p is not equal to q). Show that there is a unique…

A:

Q: 3. Show that if a and b are integers such that alb, then ak|b* for every positive integer k.

A:

Q: ) Show that if p is a prime such that m +1< p < 2m + 1, then p divides (2m+1).

A:

Q: Let a and b be non relatively prime positive integers. If there exist integers s and t with as + bt…

A:

Q: Prove that 3 and 1+v-5 are relatively prime in Q(V-5), but that there exist no integers A, µ E…

A: We will use the basic knowledge of ring theory and UFDs to answer this question properly, in detail.

Q: Let m be an integer, if m is an even integer then prove that m + 7 is odd

A:

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: m and n are relatively prime if and only if am+bn=1 for some integers a and b.

A: It is given that :am+bn=1--(1) for a,b,m,n ∈ Z Now, if  k divides both m and n, then there exist p…

Q: 12. Prove that for all integers a and m, if a and m are the lengths of the sides of a right triangle…

A:

Q: Prove that for every ponitive integer a if aº – 6° is a prime for some positive integer 6 5 = 2 or 5…

A:

Q: . Use direct proof to prove that, if x is a positive odd integer, then z – 1 is wholly dividable у…

A: x^2-1 is divisible by 4.

Q: Prove that if m and n are integers and mnis even, thenm is even or n is even.

A:

Q: If a and b are nonzero integers and m is the least common multiple of a and b, prove that (a) ∩ (b)…

A: Consider the given statement. (a) ∩ (b) = (m) Consider the left side of the equation. As, m is the…

Q: Prove that if the product of two integers qs is odd, then q2 + s2 is even. Then show that for…

A:

Q: Prove that a positive integer n is p

A: Topic :- Number system

Q: Suppose a is a positive integer and p is a prime. Prove that pla if and only if the prime…

A:

Q: there are no integers x and y such that x is a prime greater than 5 and x=6y+3

A: Given query is to check the statement whether it's possible or not.

Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZ≈ Z/mnZ if and only if m and n are…

A:

Q: Prove that if m and n are even integers, then mn+ 1-m-n is an odd integer.

A: Given :- m and n are two even integers. To prove :- mn + 1 -m -n is odd integer

Q: Show that if 2m +1 is an odd prime, then m 2" for some nonnegative integer n.

A: We solve this question by contradiction

Q: Prove directly from the definitions that for all integers the Same parity Divide Bnt 19. and E m and…

A: If m an n are both even  let m= 2a and n= 2b   then  5m+7n=5(2a)+7(2b)=10a+14b=25a+7b therefore,…

Q: Let p and q be primes. Prove that pq +1 is a square if and only if p and q are twin primes.

A:

Q: Prove that if m and n are integers and mn is even, then m is even or n is even.

A:

Q: Let m, n be positive integers. If m n, prove that 5m - 1 5" - 1

A: We show that 5m-1|5n-1 whenever m|n ,for any positive integer m,n

Q: Prove that If p is prime, p divides k^2 + m^2 and p divides m^2 + x^2, then p divides k^2 + x^2

A:

Q: Prove that if a, b and c are positive integers such that a divides b and b divides c, then a divides…

A: It is given that a,b and c are three positive integers. It is also given that a divides b. Then it…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.