4. (а) Let a, b, c E Z. Prove that, if a | b, then ac | bc, but that the converse may be false. (b) Prove that mn | a. Let a, m, n E Z. Suppose that m and n divide a and that (m, n) = 1.

Q: U is the set of the numbers on a 6-sided die and B is the set of numbers on a 6-sided die that are…

A: Given If U is the set of the numbers on a 6-sided die and B is the set of numbers on a 6-sided die…

Q: Let ū = (-1,2,4) and ū = (-3,5, 2). Compute the following: (a) ū+ū (b) .공 (c) ||해 (d) ũ - 7 (this is…

A: (a) Find u→+v→ as follows: u→+v→=-1,2,4+-3,5,2          =-1+-3,2+5,4+2          =-4,7,6

Q: Let A=(1, 0), B=(0, 1), C=(1, 1), and D=(r, 1). Verify that R(A, B, C, D) = r.

A:

Q: Let a, b c and d be real numbers. Show that ab-cd = b(a-c)+c(b-d), and deduce that |ab - cd| ≤…

A:

Q: show that }I f,lm) Ck-m) 2.

A: For, ∑m=0kf1mf2k-m, substitute k-m=n or m=k-n: ∑m=0kf1mf2k-m=∑k-n=0kf1k-nf2k-k-n…

Q: 4. Prove for A = {x: x = 12k, k = Z}, B = {x : x = 3k, k = Z}, and C = {x: x = 4k, k = Z} that A…

A:

Q: 25. Show that |v|u + Ju|v and |v|u – |u| v are orthogonal.

A: Two vectors are orthogonal if their dot product is zero. If the dot product of two vectors is zero,…

Q: 6. Prove that if alb and k E Z, then a|(b+ ak).

A:

Q: Let a, b, q, r E Z and assume that a and b are not both zero and b and r are not both zero. Show…

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Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

A:

Q: Let r ERQ. Prove that for all a, b, c EQ, =cr iff a = 0 and b = c. a + br

A: Given that r∈RQ Let a,b,c∈Qa+br=cr iff a=0 and b=c

Q: 1. Let a, b € Z so that 2a + 3b is a multiple of 11. Prove that 11 divides a - 46.

A:

Q: Let a, b, c € Z with c‡0. Prove that if a | b, then ac | be

A: Let a,b,c∈Z and c≠0. Let ab. Then b=ak, for some integer k.

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: 2. Let m, n, a, b € Z. Prove that if 6 | m and 6 | n then 3 | (am + bn +15).

A:

Q: 9. Let a, b, and d be positive integers. Prove that if (a, b) = 1 and d divides a, then (d, b) = 1.

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Q: a. Let U = { a, b, c, d, e, f, g} X= {a, c, e, g} , Y = { a, b, c }, and Z = {b, c , d, e, f} a1.…

A:

Q: If a € R such that 0 0, then show that a = 0.

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Q: . Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A2 + B2). (a) Prove the statement.…

A: 5. Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A^2 + B^2). (a) Prove the…

Q: ). Find 1 |z-j|=2 Z

A:

Q: (1) if is reflexive, symmetric, or transitive. For a, b = Z, define ab if a + b is a multiple of 3.…

A: A relation is a subset of the cartesian product A×B where A and B are non-empty sets. For cartesian…

Q: Let A={2,y,z} and B={3,x,z}. How many distinct elements are in (A×B)∩(B×A)

A: Given, A={2,y,z} B={3,x,z} How many distinct elements are in (A×B)∩(B×A) ?

Q: 26. Let A=61] Find a) A V B. and b) Α Λ Β. B = [id] c) A OB.

A:

Q: Let a, b, c E Z. Prove that if (a, b)= 1 and cl(a² + b), then (a, c) = 1.

A: Given gcd(a,b)=1  So 1 can be expressed as linear combination of a and b  That is 1=ax+by…

Q: Let a, b e Z so that 2a + 36 is a multiple of 11. Prove that 11 divides a - 4b.

A: