8. Vn E Z, if n is prime, then 2" +1 is also a prime number. 9. For every positive integer n, n2 – 2 is not prime is not necessary for n to be odd. 10. The difference of the squares of any two consecutive positive integers, is odd.

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: Based on Theorem 4.4.3 P.169, when n > 2, which of the following is biggest? 1. n! 2. n^n 3. (n!)^2…

A: We have to find biggest number among of them

Q: 1. Compute the value of (), for n > 3.

A: Formula: nr=n!r!n-r!

Q: 7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a…

A:

Q: (mp+nq)−(mq+np).

A: Given,  m-np-q=m·p+n·q-m·q+n·p It is to be proved.

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

A:

Q: 7. Suppose a1, a2,...,ɑn are integers, r is a solution to the equation 1" + a,r"-1 + a2r"-² + + a, =…

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Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

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Q: , 7/4 /213, 33 b) Solve e In(9) + 21 = 41 for æ. a) Write In(- -) in the form a lIn(7) + bln(3)…

A:

Q: If n^2+6n+11 = x^2+y^2 for positive integer n, x and y prove [xy/2] is composite. Note: [t]…

A: Given :              n2+6n+11 = x2+y2              t ≤ t

Q: 4. Prove that the difference between the square of any odd integer and the integer itself is always…

A:

Q: Which of the following alternatives provides a direct proof to show that for all n eZ. If n- 1 is…

A: Use the following concepts, to obtain the required result. An even integer m can be written as a…

Q: Q1 (a) Calculate the coefficient of æ'yz in (r+y+2)13. (b) In her discrete math class, Karen was…

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Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

Q: For every positive integer n     12+22+32+...+n2=n(n+1)(2n+1)/6 Let S(n)=12+22+32+...+n2 How do…

A: See the attachment

Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

Q: 7. 7. Suppose X - N(u.4), which is true?( X- (A) N(0.1): (B) PIX S0} = 4 (C) PIX-u>2; =1-0(1): (D)…

A: Here we use the properties of normal distribution to solve this problem.

Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: 5.43. -- ("*)(*") = ()- in (m+n+: sk=-m r+s+1

A: We have to prove given relation by using counting argument.

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: Fix k, n E N with n > 2 and k <n – 2. Prove that: | )-(".) · ) (:) ) (^) = (":") 2 + k 2 n – 2 n n +…

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Q: in is where n is an integer. e(1+4n)i where n is an integer. O equal to the above equal to the above…

A:

Q: Suppose that Z~N(0,1). Then find Pr(2.78 < Z < 2.84)

A: Given data, Z~N(0,1) That is Mean=0 Standard deviation=1 P(2.78 &lt; Z &lt; 2.84)=?

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: 5. Prove that for all n n [j³ = j=1 n² (n + 1)² 4

A:

Q: 6. Verify Euler's Theorem for n = 16 and a = 5. (7 from 6.5)

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Q: Suppose that m, n are two integers such that m3 – 2n2 = 1. Does it follow that m and n are…

A:

Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

A:

Q: Suppose X~N(165,16), and Z~N(0,1). Then x = 173, is equivalent to z = ______. (Round your answer to…

A: Given: X~N(165,16) μ=165σ2=16Thus, σ=σ2=16=4 And X=173

Q: Let m and n be positive integers and let k be the least common multipleof m and n. Show that mZ∩ nZ…

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Q: Let N=20. Show using the Sieve of Eratosthenes how to find all prime numbers between 10N and 10N+100…

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Q: of the apples is equal to 3 of the oranges. If there are 12 mor 5 In a fruit stall, - - annles than…

A: Fraction:  A small part of something.  In other words, numerical value that is not a whole number A…

Q: 18. Show that Σkind(k)³ = (Σk|n d(k))² for all positive integer n.

A:

Q: 17. prove that Edln H(n/d)7 and σ(n ) σ(n) Σ %3D

A:

Q: 4. Given that ! n+1 0.47199 determine the value of " +1

A: Here we have to find the given value.

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&gt;3.

Q: Suppose m=nd where n is odd and 1<n<m. Prove that 2m+1 is composite.

A:

Q: Suppose X~N(125,16), and Z~N(0,1). Then x = 141, is equivalent to z = ______. (Round your answer to…

A: For large sample n&gt;30 so, it is a z test. x~N(125,16) =125+16 = 141= x.

Q: Show that 6n – 1 and 12n+1 are relatively prime for any n e Z.

A:

Q: Let n ∈ Z, Is it true 4 | n4 – n? Prove it or counterexample.

A:

Q: 7 and for n > 2, an = an-1 Let, a1 = express an in terms of n.

A: For a geometric progression, the ratio of two consecutive terms is a constant. The nth term, an in a…

Q: What is the second step to prove that S, = 2+ 22 + 2' +...+ 2* = 2(2* – 1)? Show that S, is valid…

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Q: Prove that if n > 1 is a number with φ(n) = n/k, then k = 2,3.

A:

Q: Suppose X~N(163,7), and Z~N(0,1). Then x = 152, is equivalent to z = ______. (Round your answer to 2…

A: Z-score: The z-score value is a numerical measurement that describes a value of the relationship to…

Q: 8 Show that 1+2++ In S 1|+ |r2|++|rn| for any numbers r1, 22, ....

A: To show,    x1+x2+.....+xn≤x1+x2+.....xn   for any numbers  x1, x2,....., xn . Which is generalized…

Q: Suppose that m,n are two integers such that m3−2n2=1. Does it follow that m and n are relatively…

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Q: Let x and y be positive integers. When is x + y3 a prime number? When is x+y a composite number?…

A:

Q: 8. Working over R, suppose that M = PDP ' where P 1 Then, for any positive integer k, we have that…

A: To find Mk =?

Q: For n>2, let an=5n/n2-2 ,Determine for which n, |an| < 1

A: Given, an=5nn2-2|an|&lt;1 Use the given inequality for an and find the quadratic equation for n and…