Let mi EZ for i=1,2,...,k and gcd (mi, mj) = 1 for all i ‡ j Prove that for x,y e Zif x = y (mod m₂) then, where m = m1 m2 mk. x=y (mod m)

Q: 10. Let w = f(z) = Find f(1 - i) 1+z 1-z A. 1 - 2i B. −1+2i C. -1-2i D. none of the above

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A: Using the definition of modulo N.

Q: 4. Let f(x) = r3 +x² – 3x – 1. It is known that f(2) = 0(mod 5). (i). Find a solution to f(r) =…

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Q: 35. Prove: If xn = :o(an), then xn = = O(an). Show that the converse is not true.

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Q: Let f₁=1,₂ = 1 and for all neN,n23,f=f+f2. Prove 1+√5 1-√√5 2 2 that for all neN, S √5

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Q: The formula for slope between two points (x1, y1) and (x2, y2) is: m = 5-4 O True O False

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Q: Wi <² -- = K (K+1)(2K+1) G

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Q: 6. Prove that if gcd(a, b) 1, then gcd(a +b, ab) = 1. %3D %3D

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Q: Suppose x, n ∈ N and n not equal to  0. Then, show that xn is positive if and only if x is positive

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Q: [2 1] 1 3 l3 4] 4 Q2) Assume A %3D and B show that (AB)' = B' A' 6. %3D

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Q: 2. Prove that Qn [2, 3] ~ N.

A: Two sets A and B have the same cardinality if there exists a bijection (onto , one-to-one…