Prove the following statements: b. Let a, b E Z. Then a = gcd (a, b) if and only if alb. c. Let a, b, c € Z. If albc and gcd (a, b) = 1 then a/c.

Q: Let a, b, c E N. (a) Prove that if gcd(a, c) = 1 and gcd(b, c) = 1, then gcd(ab, c) = 1. (b) Prove…

A: a, b, c∈ℕ

Q: Let a, b, x, y ∈Zbe such that ax+by= 11. May we conclude that gcd(a, b) = 11?Why or why not? (Write…

A: Consider the given information:Let a,b,x,y∈ℤ be such that ax+ by=11.

Q: -8. a. Show that if A and B are independent, then P(A|B) = P(A). b. Say we have sets A, B, and C (do…

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Q: 3. Prove that if d is a common divisor of a and b, then d = gcd(a, b) if and only ir gcd(a/d, b/d) =…

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Q: 2. Let f: ZxZ-ZxZ be defined by f(m, n) = (3m+n,n²) a. Is fone-to-one? b. Is fonto?

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Q: Prove that if A – B – D and B – C – D, then A -C– D.

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Q: Let ab € [0,00) e with a² ≤ b² Show that a ≤ b

A: The objective of the question is to prove that if a and b are real numbers such that a^2

Q: Let a, b, n E Z. Prove: If a|n and b|n with gcd(a, b) = 1, then ab|n. (I

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Q: If a and b are even then GCD(a,b) must be 2

A: Given statement :- if a and b are even then GCD(a,b) must be 2

Q: Let U {1,2,..., 100}, and let A consist of 15 unknown numbers taken ran- %3D domly from U. Let B…

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Q: Ema thinks that (u) xw, exists. Paul disagrees with her. Who is right? Explain why.

A: Dot product gives a scalar value  .

Q: Suppose that a divides b and that c divides b. Suppose that gcd(a, c) = 1. Prove that ac divides b.

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Q: Prove that |A U B| + |A n B| = |A| + |B|

A: Let A and B are two sets. A = number of elements in the set A. B = number of elements in the set B.…

Q: Assuming that gcd(a,b)=1, prove the following (a)  gcd(a+ b,a - b)=1 or 2. [Hint: Let d=gcd(a+ b,a…

A: Let d=gcd(a+ b,a - b)    => d | (a + b) and d | (a - b)     => d | {(a + b) + (a - b)} and d |…

Q: 4. Let A

A: Given,                A=1, 2, a, b, ϕ

Q: Let a, b, c E Z. show that ca & cb if and only if c (a, b).

A: Division properties are used to prove it

Q: Let a, b € Z. Prove that if a ≥ 11, then af b or ał (b+7)

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Q: Let S = {a, b, c, d, e, f} with P(b) = 0.21, P(c) = 0.25, P(d) = 0.2, P(e) = 0.1, and P(f) = 0.11.…

A: