If p,q,r are prime number but p is not equal to q, then what is gcd(p,q)? If p|rq, prove that p|r and deduce that p = r.

Q: 22] calculate the number of combination and permutation of, 5 distinct objects taken 2 at time?

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Q: Let U = {1,2,3,4,5), A = {1,3,5}, B = {1,3,4), and C = {2,3,5). (a) Find A U (BUC). (b) Find (AUB)…

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Q: If X ~ N(5,2). Calculate P(X<6) [Draw a figure]

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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: Use a Fourier sine expansion to solve ĉu ô’u ôt ôx? u(0,t) = u(1,1) =0,1> 0, = x sint in (0,1) ×…

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Q: Suppose P(A) P= 0.3, P(B) = 0.4 and P(A ∩ B) = 0.12. Which of the following is Correct? A) P(A U B)…

A: The probabilities of the events are given as follows:

Q: Show that If (A ∩ B = ø) then P (A ∪ B) = P (A) + P (B)

A: Given that A ∩ B = ø This means P(A ∩ B ) = 0

Q: Let U=​{1,​2, 3,​ ...,2400​}. Let S be the subset of the numbers in U that are multiples of 3​, and…

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Q: 1. Let P(A) = 0.3 and P(B) = 0.6. a) Find P(AUB) when A and B are independent. b) Find P(A/B) when A…

A: P(A) = 0.3P(B) = 0.6

Q: Suppose Z∼N(0,1). Then P(2 ≤ Z ≤ 3) is the same as

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Q: prove the ged (atb,a?+b³) =16Y2 assuming that gcd (a,b) =1 •

A: It is given that gcd(a,b)=1

Q: 12 y 2 4xy³z² 4877

A: formula used  xa×xb=xa+blet  A=12x4y8z74xy3z2

Q: Consider the following. Find Av. Av = A = 10-1 1 1 1 = 7 -1 Complete the following statement. Since…

A: Given that A is a 2X3 matrix and v is 3X1 vector so product is possible. The null space of A,…

Q: Problem #7: Consider the following two elementary matrices. --[0-61 E₂ = 9 Problem #7: E1 = Suppose…

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Q: Show that for integers a, n satisfying 0 < a <n, ged(a, n) = gcd(n, n-a)

A: Solution is given below:

Q: Prove that if A and B En and Ac B, then

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Q: Let p, q, and r be prime natural numbers with p # q and n= p2q. Then, all the subfields of GF(r")…

A: Answer: Option (C) Given that, p,qand r are prime natural numbers and n is positive integer. Let F…

Q: are 51 software students taking Let pi, p2,.. Psi be their final cam scores. Hence, cach p, is an…

A: Given that p1, p2, p3,..., p51 are the final exam scores of 51 software students who are taking MATH…

Q: 5) Suppose that W1 < R" and W2 4 R". Prove (W1 n W2) < R"

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Q: fo = 1, fi = 26, 7fn = fn+1 + 10fn 1 The following four questions are all based on the paragraph…

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Q: Let f₁=1,₂ = 1 and for all neN,n23,f=f+f2. Prove 1+√5 1-√√5 2 2 that for all neN, S √5

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Q: 3. f(x) %3D — 5x? + 15х — 11

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Q: Then, there exists k1≠0, k2≠0  such that a,b=k1c,k1d   and  c,d=k2e,k2f. a,b=k1c,d=k1k2e,…

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Q: 1 Show that n2 + 1 IN V

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Q: 3) Let O# a ER". Then that a' ERTI prove

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Q: Prove that 1.3 + 5.7 + 9.11 + =

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Q: Prove this statement: we have numbers {a1, a2, a3 .,a2n} which are even and positive, but 0 < {a1,…

A: As, we know that, A.M.≥G.M. ---(1) A.M. of numbers, a1,a2,a3...a2n=a1+a2+a3+a4...+a2nn G.M. of…

Q: 6x:-ld 212CK) 's aving of integer num 12 and fo), gaJE Z, ) such That 9Cx) = Prove Fhat dey (fagx))…

A: Given  Ring Z12(x)  f(x) = 2+5x+4x4  g(x) = 1+x+x2+3x3    So clearly,  degree of f(x) = 4  degree of…

Q: Queshon 6) Compute the arthogonal projection of 8 onto the Line L through -S] and the origin. -6 LS.

A: We have to find the orthogonal projection of v on the line through P and the origin.

Q: If P, stands for "P., then prove that I+1. P, +2. P, +3. P, +...+n. P, = (n+1)!.

A: that is will be...equal the 1 + 1. 1P1 + 2. 2P2 + 3. 3P3 + …. n. nPn = n+1Pn+1. and we know that nPr…

Q: let it be T=11,2,3 } S:la,b,c] Proved P(TNS)E P(S)NP(T)

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Q: Prove that Z(D,) # {1} if and only if 2|n (hint: if 2|n look at ri).

A: Let us suppose, Dn=e,r,r2,..,n-1,f,rf,..,rn-1f and nis divided  by 2. Then, n2 is a integer. We see…

Q: If gcd(m,p) = 1 and p is prime, show that gcd(m,pk ) = 1 for all k≥ 1.

A: To show, If gcd(m, p)=1 and p is prime, show that gcd(m, pk) = 1 for all k≥1.

Q: Supposernc N and n 4 0 Then show that rn is positive if and only if r is positive

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Q: (14) Let E denote the set of even integers, O denote the set of odd integers, and x, y € Z be any…

A: I have given the answer in the next step. Hope you understand that

Q: Suppose X~N(112,16), and Z~N(0,1). Then x = 112, is equivalent to z = ______.

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Q: Let a1 = a2 = 1 and suppose an = an-1 + an-2 for n ≥ 3. Prove that a1a2 + a2a3 + · · · + a2n-1a2n =…

A: We have given that , a1 = a2 = 1 Also , an = an-1 + an-2 We need to prove that ,  a1a2 + a2a3 +…

Q: Suppose X~N(113,18), and Z~N(0,1). Then x = 94, is equivalent to z = ______.

A: Given that, Suppose X~N(113,18), and Z~N(0,1). Here, mean μ=113 and standard deviation σ=18

Q: Prove the soquen enat 2efin ea For n EIN Conv e

A: Introduction: When each term of a sequence is greater than its previous term, the sequence is…