7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a prime number, then m must be composite.

Q: 1. If N = 12, k = 8, and n = 5. Find P(2). %3D

A: X follows a hypergeometric distribution with N=12, K=8 and n=5 and k=2 We will use the formula:…

Q: Let m, n >0 be integers. Prove that if m +n 2 29 then m > 15 Question 11 or n> 15 by contradiction.

A: To prove m+n≥29 then m≥15 or n≥15 where m,n≥0 be integers.

Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

A: .

Q: 26. In how many ways can up to 4 students be selected from 6 girls and 5 boys if each selection must…

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Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: 3. Show that if n = 3 mod 4, then n cannot be the sum of squares of two integers.(i.e there are no…

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Q: Suppose a set of 8 numbers are selected from the set: {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13;…

A: It is given that 8 numbers are selected from the set 1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: 1. Will there be at least two people who celebrate their birthday on the same day Yes, definitely O…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: 4. Prove that if m and n are positive integers and x is a real number, then %3D m

A: We know ⨽x⨼ = x if x is an integer. Here, since m and n are integers, we have the following steps:…

Q: 2. Suppose a, b, c, d are integers and that (a, b) = (c,d) = 1. If (g) + (á) show that b = ±d. is an…

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Q: 1. Let n E N. Prove that n and nº have the same last digit.

A: See the attachment

Q: Disprove the claim: 3 m E N such that m > 3 and m? – 1 is a prime number.

A: The claim is given : ∃ m∈N such that m≥3 and m2-1 is a prime number Now, m2-1=(m-1)(m+1)--(1) Case…

Q: 2.4 the humbe OF Counties in State A and the number OF COnties in SEGLe B even integers whose sum is…

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Q: contradiction

A: Please refer to the image below

Q: 9. (Chapter 10, #20) Prove that for every n E N, (1+2+3+ · ..+n)2 = 13 + 23 + 33 + ...+ n³.

A: To prove the result ''∀n∈N,1+2+3+...+n2=13+23+33+...+n3'' by Mathematical induction: First prove for…

Q: We need to attempt to find an exact value of n such that  P(r ≥ 6) = 0.98. Clearly to guarantee…

A: We have to fill blanks from given statement Clearly to guarantee at least six people are literate,…

Q: Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p ≥ 5, at least one of…

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Q: The system x E 3 mod 9, x E 2 mod 4, has O A. no solution. O B. Unique solution. O C. finitely many…

A: Introduction: The Chinese remainder theorem is very significant in solving a system of linear…

Q: If in a AP the sum of m tern=ms is equal to n and the sum of n terms is equal to n ,then prove that…

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Q: 9. Show that if p is an odd prime and p = 3 mod 4 then p is not the sum of the squares of two…

A: In the given question we have to prove that if p is an odd prime and…

Q: If a and b are relatively prime then alb

A: Given below the detailed solution

Q: Prove directly from the definitions that for all integers m and n, if m and n have the same parity,…

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Q: 3. Prove that when you select n+1 numbers from {1,2,...2n} two of them Must be consecutive (that is,…

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Q: Prove that when you select n +1 numbers from {1,2, ... 2n} two of them must be consecutive (that is,…

A: Use inductive argument and Pigeon hole principles

Q: Let x, y E N, determine each of following statemen is true or false (N means Natural number, natural…

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Q: prove by contrapositive : for all n,m integers if nm is even then n is even or m is even

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Q: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

Q: Prove that if 10|n, then 10|n^2+n

A: Prove that if 10|n, then 10|n^2+n

Q: Theorem 11.3. If p and q = 2p+1: primes, then either q | M, or q| M,+2, but are not both.

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

Q: n ΔMNO MO is extended through point O to point P, m∠NOP=(7x−17)∘, m∠MNO=(x+20)∘, and m∠OMN=(3x−7)∘.…

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Q: If gcd(a, b) = d, then a and- are not relatively prime. d %3D

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Q: 2. Prove that if m and n are integers and mn is even, then m is even or n is even.

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Q: Disprove the claim: 3 m E N such that m 2 3 and m? – 1 is a prime number. -

A: Given that ∃m∈ℕ such that m≥3 and m2-1 is a prime number. Take m=4. It is known that 4∈N And 4>3.

Q: * 10. Is the following proposition true or false? Justify your conclusion. For each integer n, n is…

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Q: Prove that if m and n are even integers, then mn+ 1-m-n is an odd integer.

A: Given :- m and n are two even integers. To prove :- mn + 1 -m -n is odd integer

Q: Prove directly from the definitions that for all integers the Same parity Divide Bnt 19. and E m and…

A: If m an n are both even  let m= 2a and n= 2b   then  5m+7n=5(2a)+7(2b)=10a+14b=25a+7b therefore,…

Q: Let p and q be prime numbers. (1) If p is not equal to q, how many generators does Zpq have? (2) If…

A: Given p and q be primes.

Q: 5. Let C = {x ∈ Z|x ≡ 7( mod 9)} and D = {x ∈ Z|x ≡ 1( mod 3)}. (a) If C ⊆ D? Justify your…

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Q: 1. Given a deck of 30 cards labelled with numbers 1- 10 (with 3 cards of each number), how many…

A: From the given information,  A deck of 30 cards labelled with numbers 1-10 (with 3 cards of each…

Q: Exercise 9.3.2. Show that if p and q are primes, then the number of positive integers less than pq…

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Q: Since the next prime after 7 is​ 11, the smallest composite number whose prime factors are all…

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Q: Show that the integers m = 3k . 568 and n = 3k . 638, with k E Z>o, satisfy simultaneously $(m) =…

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Q: Prove that if m and n are integers and mn is even, then m is even or n is even.

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Q: 4. Let bo = 1, bị = 3, and for each integer k > 2, we have bk = 4bk–1 – 4bk–2 Prove that for all…

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Q: Prove that If p is prime, p divides k^2 + m^2 and p divides m^2 + x^2, then p divides k^2 + x^2

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Q: if m is even and n is odd demonstrate that m*n= is even

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Q: 4, Suppose P(A) 2 0.8 and P(B) > 0.5. Prove P(AN B) > 0.3. Hint: You may use P(AUB)< 1.

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