Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

Q: Use a direct proof to show that if a and b are positive integers, then ab+ba22.

A: Answer

Q: Prove the following statements: iii. Suppose that a, b, c and d are positive integers, prove that…

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Q: The question is Prove or disprove: for all integers a, b, c, d, if a divides both b − 2c and c − d,…

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Q: Let a and d be integers. Prove that if d|(8a+7)and d|(5a+4), then d|3

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Q: Let a, b be nonzero integers. Prove that (a, b) = 1 if and only if (a + b, ab) = 1.

A: Given  Let a, b be nonzero integers  To Prove that gcd (a, b) = 1 if and only if gcd(a + b, ab) = 1…

Q: = Let a, b, and d be positive integers. Prove that if (a, b) : divides a, then (d, b) = 1.

A: Definition of gcd (a, b) =1 is a and b does not have common prime divisor. Using this we can easily…

Q: Let a, b, c be positive integers and let d = (a, b). Since d|a, there is an integer h with a = dh.…

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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: 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: " Show that the integers m = 3* . 568 and n = 3k. 638, where k > 0, satisfy simultaneously t(m) =…

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Q: Show that if a1,: a2., an are integers that are not all 0 and c is a positive integers, then…

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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, a1 = 3 and for n > 2, an = 2an-1+ 1, express an in terms of n.

A: a1 = 3 and, an = 2an-1 + 1 for, n≥2 We have to find the generalized term using the recursion…

Q: Let a, b eN. Prove that if a + b is even, then there exists nonnegative integers x and y such hat x²…

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Q: •Prove: For all integers a, b, and c, if a|b and alc, then a|(b+ c). we can assume a #0.)

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Q: Show via proof that for two positive integers a, b where a 2 b, (use combinations e.g. n choose k)…

A: We shall prove for the given statement. Though it has little ambiguities.

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: If a, b and c≠0 are integers, then a|b if and only if ac|bc.

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

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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: Find a pair of integers x and y such that 66x + 48y = gcd(66, 48)

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Q: Let a, b and m be integers. Prove that if 2a + 3b ≥ 12m + 1, then a ≥ 3m + 1 or b ≥ 3m + 1.

A: Here I have corrected the statement of the Question and provide a detailed solution of the problem.

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

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Q: Prove by contradiction that if 3x? 2 is irrational then r is irrational.

A: We have to show that x is irrational if 3x2-25 is irrational. We have to prove the statement by…

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

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Q: For all positive integers a, b, where a < b, if gcd(a, b) = 1, then a b. O True O False

A: " Since you have asked multiple questions, we will solve the first question for you. If you want any…

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: Suppose that A and B are two positive numbers and that x = In(A) and y = In(B). Write the following…

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Q: For positive integers a and b, show that a ab

A: Consider the provided question, For positive integers a and b,  we need to show that, xab=xab

Q: Leta b,c be integers, no two of which are zero and d = gcd(a,b,c). Show that d= ged(ged(a,b).c)…

A: Let d=gcd(a,b,c). Show that:  d = gcd(gcd(a,b),c) = gcd(a,gcd(b,c)) = gcd(gcd(a,c),b).

Q: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

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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: For all positive integers a, b, where a < b, if a {b, then gcd(a, b) = 1. O True False

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Q: If a, b, c are integer powers of 2, prove that equation a° + 64 .3 + 64 = c5

A: Given: Let's take a,b,c are integer powers of 2. Prove that a3+b4 =c5  does not have a solution.

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: Let a and b be integers which are not both zero, and let d= gcd(a, b). Prove that gcd (a/d,b/d)= 1.

A: Given that, a and b be integers and in both of them at least one is non-zero. And, greatest common…

Q: (b) If r23= 1, prove that R² (23) = 112=113 and 01.23 = 01 (1-12).

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Q: Let a and b be positive integers and let d = gcd(a, b) and m =lcm(a, b). If t divides both a and b,…

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Q: prove if a and b are both odd integer, then 16/ (a^4) + (b^4) -2.

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Q: Let A ={ x∈Z | x=5a+2 for some integer a }, B ={y∈Z | y=10b-3 for some integer b }, C={ z∈Z |…

A: Let A ={ x∈Z | x=5a+2 for some integer a }, B ={y∈Z | y=10b-3 for some integer b }, C={ z∈Z |…

Q: Suppose a, b, and c are integers and suppose a b and ałc, where atc denotes that c is not divisible…

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Q: Suppose a, b, c and d are integers. Prove that  if a divides b and c divides d, then ac divides bd

A: According to the given information:   It is given that a divides b, there is an integer x such that…

Q: such

A: Given: 165x2-21y2=19

Q: tet r and m be integer, prove that 17/20+3m iff, 17/9n

A: Solution

Q: If a and b are two integers such that a = bq +r, with b> 0, then %3D gcd(a, b) # gcd(b,r) gcd(a, b)…

A: Let a = bq + r   … (1)Now,  let d1   = gcd ( a,b) then ⇒d1  | a  and   d1  | b ⇒d1 | a - bq   and d1…

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Q: Prove that if p,q, and r are integers, r|p and r|q, then r^2|pq.

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Q: For any non negative integer n , prove that an – bn is divisible by a-b,where a and b are any…

A: Use the binomial factorisation

Q: Prove that if m is a positive integer and x is a real number, then [m2] = [=] + z++ + 2 + |x + т — 1…

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