6. Prove that for all integers a, b, and c, if a² + b² = c², then at least one of a and b is even.

Q: How do I prove that if a, b, and c are integers with c not equal to 0 then ac = bc => a = b?

A:

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

A:

Q: 23. Prove that for any integer a, 9/ (a² – 3). -

A: Prove that for any integer a, 9 ⫮ (a2 – 3).

Q: 2. Prove by mathematical induCHon that १) के) 22 32 42 2n is Hrue For all integers nz 2

A:

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

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

Q: Prove: For integers m, n, 12mn – 9 is an odd integer.

A:

Q: Let a, b be two positive integers with (a, b) = 1. If ab = c, then prove that there exists

A:

Q: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

Q: 2. Prove: If a, b, c, and d are integers such that a + b =c+d, then a A=

A:

Q: (2) Show that for any integers a,b we always have a? – 4b # 2. -

A:

Q: Prove that if a > 2 and b are integers, then a † b or a { (b+1).

A: Given that a≥2 and b are integers. We will prove it by contradiction. Let a|b and a|(b+1) then there…

Q: Prove that if x is an integer, then x2 has the same parity as x

A: Parity means that if the integer given is odd or even, the given function will be similarly odd or…

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: 6. Use contradiction to prove that, for all integers k > 1, 2/k+1+ 2/K+2

A:

Q: Show that for all n, m∈Z if n+m is even, then either n and m are both even or n and m are both odd.

A: Use the results of addition of integers.

Q: 4. Prove that if m and n are positive integers and x is a real number, then %3D m

A: We know ⨽x⨼ = x if x is an integer. Here, since m and n are integers, we have the following steps:…

Q: 3. Prove by contradiction, that if x and y are odd integers, then there is no integers z such that…

A:

Q: Prove that if n divides a, b, and c, then n³ divides a²(b – c) + b² (a – c) +²(b – a).

A: A product is a multiplication of two or more terms and these terms are called the factors of the…

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: Let a and b be two integers. Prove that if a+b is even then a-b is even.

A: Given that a and b are any two integers. We are trying to prove "if a+b is even then a-b is even."

Q: Prove that If k, q, w and r are integers, and if c|k and c|q, then c|(kw + qr)

A: We know that  Definition of Divisibility : If a and b are integers such that a≠0, then we say "a…

Q: Let a, b be nonzero integers with (a, b) = 1. Compute (a + b, a – b).

A:

Q: Show that is a and b are positive integers such that (a | b) and (b | a), then a = b

A:

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: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

A: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

Q: Prove directly from the definitions that for all integers m and n, if m and n have the same parity,…

A:

Q: Suppose that all the entries in A are integers and det A = 1. Explain why all the entries in A–1 are…

A: Inverse of a matrix A is calculated by,

Q: Prove that there are no integers m and n such that m2=4n+2.

A:

Q: Suppose we randomly select a subset of positive integers. How large must the subset be, if we want…

A: According to the given question, if any positive integer n belongs to the required subset then it…

Q: x]+n x+n m m

A: By using definition of greatest integer function, For x∈ℝ we have, x=x+x, 0≤x<1 where, x is…

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 by contradiction that for all integers a, b, and c, if a|b and a | c, then a |(b + c).

A:

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: 4) Determine if the following statement is true and either prove it or provide a counter example:…

A:

Q: Prove that if ∣A∣ = 1 and all entries of A are integers, then all entries of ∣A−1∣ must also be…

A:

Q: If a, b, and c are integers such that b is a multiple of a and e is a multiple of b², then c is a…

A:

Q: (1) Disprove the statement "For all integers a, b if a | b2 and a < b then a | b." by giving a…

A:

Q: 2. Prove that if m and n are integers and mn is even, then m is even or n is even.

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: Prove by contrapositive: For all integers a,b and c, if a [ bc then a [ b. (J means does not divide)

A:

Q: Let a, b, c be non-zero integers. Prove that gcd(a, bc) = 1 if and only if gcd(a, b) = 1 and gcd(a,…

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: Suppose we randomly select a subset of positive integers. How large must the subset be, if we want…

A: The Pigeonhole Principle:  If there are p+1 or more pigeons, need to distribute among p pigeonholes,…

Q: a2 − b2is divisible by 5.

A: First observe that one's place of the square of a positive number can only one among 0,1,4,9,6, and…

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

A: We have to solve given problem: