Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

Q: Prove that if a and b are integers with a 0 and where + b - a = 0, x is a positive integer such that…

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Q: Show that every positive integer is of the form 2q,and that every positive odd integer is of the…

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

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

Q: Prove that there exists an odd integer m such that every odd integer n with n greater than or equal…

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Q: Prove: there exist no integer a and b for which 21a + 30b = 1.

A: We can prove this by contradiction. Take the negation of the statement and check if it is true, if…

Q: Prove that the cube of any integer has to be exactly one of these forms: 9k, 9k + 1, or 9k + 8 for…

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Q: Prove that 2 is a factor of n2 + 5n for all positive integers n.

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: let b be a nonzero integer. prove that gcd (0,b) = |b|.

A: We need to prove that for a nonzero integer b,

Q: Prove that there are no positive integers a and b such that a + b³ = 250.

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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: Prove that there do not exist three consecutive integer values of nn for which 43n+1443n+14 is a…

A: Given 43n+14 We have to Prove that  ∄  three consecutive integer values of n for which 43n+14 is a…

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

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: Let a and b be positive even integers. Prove that (a, by 2

A: Here, a,b denotes the gcd of a and b. a2,b2 denotes the gcd of a2 and b2. Given that a and b are…

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 non negative integer n , prove that an − bn is divisible by a−b,where a and b are any…

A: answer

Q: Let x and y be integers. Prove that if x2 + y2 is odd, then x+y is odd

A: Given x and y be an integers. We have to prove that if x2+y2 is odd then x+y is odd. Since given…

Q: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

Q: Prove that there do not exist integers m and n such that 12m + 15n = 1.

A: Proof by contradiction. Assume the given statement is true.

Q: snip

A: x+c is a factor of xn+cn if n≥1 is an odd integer.

Q: Prove: For all integers a and b, if a + b is odd then exactly one of the integers, a or b, is odd.

A: (1) We have to prove that a+b is odd then exactly one of the integer a or b is odd. Suppose that a…

Q: Prove that n2 – 7n + 12 is nonnegative whenever n is an integer with n > 3.

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Q: Prove that the fourth power of any integer has the form 4k or 4k + 1 for some integer k.

A: We have to prove the following -  The fourth power of any integer has the form 4k or 4k+1 for some…

Q: Prove that if a and b are integers of the form 4n+1, then ab is of the form 4n+1.

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Q: Prove that if m is a positive integer and r is a real number, then Lma) = l=] + -+ + + [-+ + r+ m

A: We are given a Inequality involving the sum of floor functions, and we have to prove that.

Q: Use the contradiction method to prove: There exist no integers a and b for which 21a + 30b = 1

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Q: If a and b are distinct integers prove that a-b is a factor of an -b" whenever n is positive integer

A: if a and b are distinct integers prove that a-b is a factor of a^n - b^n whenever n is positive…

Q: Prove: that if a is an odd integer, then 4 is a factor of a^2 -1

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Q: Let k and n be two positive integers, and assume that n is odd. Prove that there exists an integer a…

A: Given: Let k and n be two positive integers in that, we assume if n is odd.

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

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Q: Prove that if x is an even integer, thenx^2−6x+ 3is odd

A: Any integer of the form 2m+1 is odd.

Q: Prove that the statement is true for every positive integer n.

A: We have, Pn=a2n-1+b2n-1 Let us consider the statement Pn is true for ∀n∈ℕ. Therefore,…

Q: a. Prove that for every integer a, if a³ is even then a is even. b. Prove that 2 is irrational.

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Q: Prove that every integer greater than 27 can be written as 5a + 8b, where a, b e Z+.

A: Let P(n) be the statement, "For n∈ℕ there exist a, b>0 such that              n+27=a·5+b·8 Use…

Q: prove that if a and b are both odd integers, then ab+1 is an even integer

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

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Q: Prove that 3, 5, and 7 are the only three consecutive odd integersthat are prime.

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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: Prove that 3a^1 is even if and only if a is an odd integer.

A: In the given question we have to prove that Prove 3a^1 is even if and only if a is an odd integer.

Q: Prove {12a + 4b : a and b are integers} = {4c : c is an integer}

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Q: Show that any integer of the form 6k + 5 is also of the form 3j + 2, but not conversely.

A: 2. We can write 6k+5 = 6k+3+2=3(2k+1)+2 lets write 2k+1=j hence 6k+5=3j+2

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

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Q: 18. a. Prove that for every integer a, if a' is even then a is even. b. Prove that V2 is irrational.

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Q: and

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Q: Prove that for any positive integer n, Vn is either an integer or irrational.

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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…

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:

Q: Prove that there exist integers m and n so that 2m+7n=1.

A: The  integer is a whole mumber that may be positive number or negative number or zero.But not a…