Let a , b >= 1 and x ∈ Z. Prove that if xb ≡ 1 (mod a) then gcd(x, a) = 1

Q: aslal. llal-1b||<la + b| < lal + |b|

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A: Definition : Greatest common divisor ( gcd): Let  a and b be are two  integers, not both 0.  A…

Q: Assuming that gcd(a , b) = 1, prove the following: (b) gcd(2a +b, a +2b) = 1 or 3.

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Q: Prove: If a &gt; 0, then (1/a) &gt; 0.

A: It is given that a&gt; 0.It is known that if ab &gt; 0 then a &gt; 0 or b &gt; 0 or a&lt; 0 or b…

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Q: Prove that: (5a + 6b|a, b = Z} = Z

A: To prove that .Concepts used:Consider two sets  and . To prove that , we first need to show and…

Q: Let S= {n | n=5 k+ 4 for some integer k} Let T= {n|n=5 j– 1 for some integer j}

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Q: Define g: ZX Z→ Z, g (x, y) = z, where z mod 3 = x and z div 3 = y. Find g (2, 2).

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Q: Let a and b be any real numbers. Prove that Cov (ax, by) = abCov (x, y).

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Q: xʻy Let f (x, y) = if (x, y) # (0,0) and f (0,0) = 0. (x4+6y8) r4.

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Q: (b) If m < a1 + ... + an < M n, and v, is positive and + anVn < Mv1. + anUn| < Mvi Vn. decreasing,…

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Q: Show that gcd(a,n)=1 if these exist m and k such that ma+nk = 1.

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Q: Suppose gcd(a, b) = 1. Prove that ab | c if, and only if a|c and b| c.

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Q: Prove if a e G, o(a) =n, then a = reN GCD(n,r)'

A: Given below the detailed solution

Q: Show that if x is in both W and W-, then x = 0.

A: Dot product of perpendicular vectors is 0.

Q: Let X={x=(x1, x2, X3, X4) : Xi E {1,2,3,4,5,6,7,8,9,10,11,12} for i=1,2,3,4 such that x1 < x2 < X3 <…

A: Given:X={x=x1, x2, x3, x4 : xi∈1,2,3,4,5,6,7,8,9,10,11,12for i=1,2,3,4 such that…

Q: Let x € Z. Prove that if 2 | (x²-5) then 4 | (x²-5).

A: Given that .The objective, if then prove that .

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: If [a] = [1] in Z_n, prove that gcd(a, n) = 1. Show by example that the converse may be false.

A: The given data is: a=1 in Zn To prove gcd(a, n)=1

Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

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Q: Let x ∈ R. Prove that if x3 − x > 0 then x > −1.

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Q: Let a, b ∈ Z, not both zero. Prove gcd(−a, b) = gcd(a,−b).

A: We have gcd(-a , b)=gcd(a,-b) let gcd(-a , b)=d Where -a=de  , b=df  ⇒a=d(-e) , -b=d(-f)⇒d is…

Q: Find a number z (except 0) such that Az³ + Bz²+ Cz = D 2 1 3 0 -2 -4 00 ^-[ ] ³-R2 :]-[+]-[1] B= 1 2…

A: Solution

Q: Find all units of the integral domain of Gaussian integers.

A: We have to find all units of the integral domain of Gaussian integers.

Q: Let J₂ = (0, 1, 2, 3, 4), and define G: JgJgJg x Jg as follows. For each (a, b) € 7₂ × 7₂. G#, 6) =…

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