Show that C(p,1), C(p,2),..., C(p,p-1) are all divisible by p if and only if p is a prime.

Q: 3. Show by mathematical induction that, for all n2 0, the equality +++.+ = 2 – +2 holds. ... 2Th

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Q: 1. If p>9 2 5 and p and q are both primes, prove that 24 p - q.

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Q: Prove the following: (a) If a, b,c € Z+ and a+b=c then (c, [a, b]) = (a, b).

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Let m be the least common multiple of a and b. Prove that if p is a prime divisor of m, then either…

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Q: 10. If j and k are both prime numbers, and j> k, which of the following statements must be true? I.…

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Q: The greatest common divisor (gcd) of non-zero integers x and y is the the smallest positive integer…

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Q: 6. Let a1, a2,...,an be distinct positive in not containing s = a₁ + a₂ + + an. A grassh A AL icht…

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Q: 2. U={x|x is an integer, 2<x<10}. A and B are subsets of U such that A= {multiples of 3}, B={factors…

A: Using set theory we have to answer following.

Q: The permutations p = (3 5 4), q= (1 5 2 3) and r = (1 5 2) are elements of S5. pal, m. (a) Find each…

A: (a) To find each of the given permutations in cycle form, we will apply the permutations…

Q: Let [A1, A2], [B1, B2], [C1, C2] be 3 married couples who attended a concert.  In how many ways can…

A: Permutation: A permutation is an arrangement of objects in a definite order. Let n is the total…

Q: If A ∩ B = ∅ Find P(A) ∩ P(B)?] With all the steps

A: Power set of A is the set of all subsets of A Since it is given that A∩B=∅ which means there is no…

Q: In each of (a)-(f), answer the following questions: Is AS B? Is B C A? Is either A or B a proper…

A: As per the policy of bartleby the provisions is to solve only one 3 subparts of questions at a time

Q: Prove by contraposition that for all integers n, if n is odd, then 5n 4 is odd.

A: We need to prove if n is odd, then 5n - 4 is odd by the method of contraposition.  The…

Q: The coefficient of x³ in (1 + x² + x² + x6)4 is determined by the 4-tuples o integers (a1, a2, a3,…

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Q: Prove that for any n E N, nº + 2n is divisible by 3.

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Q: Prove that 17Z ∪ 7Z is countabl

A: A set X is said to be countable if there exists a bijection between f: X -&gt; N, where N is the set…

Q: Prove that for all x ∈ N, x is divisible by 4 if and only if the number given by the last 2 digits…

A: Soln) Let x be a natural number. Then we can write x as: x = x0+10x1+⋯+10nxn Since 102=4⋅25 ≡ 0 mod…

Q: 3. We say that an integer c is a common divisor of integers a and b if cla and cb. We say c is the…

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Q: Let P(n) be the statement that 1² +2²+...+ n² = n(n+ 1)(2n + 1)/6 for the positive integer n. a)…

A: Let the statement is  To Find:a) What is the statement P(1)?b) Show P(1) is true.c) What is the…

Q: a)Show by one combinatorial method that: 2 2 n C) - () - (") 2n +... + 2 n

A: Given information: The following identity is provided: n02+n12+n22+................+nn2=2nn It is…

Q: Prove that 36n is not O(35n)

A: Let, 36n∈O35n

Q: m and n are relatively prime if and only if am+bn=1 for some integers a and b.

A: It is given that :am+bn=1--(1) for a,b,m,n ∈ Z Now, if  k divides both m and n, then there exist p…

Q: justify each answer. (a) The number 1003 is prime because it is not divisible by 2, 3, 5, 7, 11. (b)…

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Q: 4) Let n 2 2 be an integer, such that 2" – 1 is prime number (e.g. if n = 7 then 2" – 1 = 127 is a…

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Q: An organization wants to assemble a cluster of n computers using wires and hubs. Each must have…

A: Solution of the problem as follows

Q: Let U = {x: x is an integer and 2 7}, B = {x: x² > 20}. ii. A = {x: x2 - 3x + 2 = 0}, B = {x:x + 7…

A: We have to determine which condition applies on set A and B.

Q: (5) (Extra credit, 5 points) Show that 102n-1 +1 is divisible by 11 (namely, 102n−1 +1 = 11 ·an,…

A: Step 1: Step 2: Step 3: Step 4:

Q: 5. Find all integers n such that Z6 x Z10 x Z4 has an element of order n.

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Q: finitely many prinies of the form in 4n  +3.

A: Suppose there are only finitely many primes and let p be the largest one. Define q by…