11- P, X Pz X P, ... X Pn, where M= Pr. Pn are distind primes Prove by inducton, For all Ppositive intsers n, if ged ca, M)=1 ,then (Pa-)CP;-=)-CPa-) =1 (mod M) a E1 (mod M)

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

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Q: 3. Find integers x, y € [0,51] such that (a) 23x= 1(mod52) (b) 23y = 4(mod52)

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Q: let x  be a fixed but arbitrary integer. Do there exist infinity many integer y such that,…

A: Let x = x0 be fixed given integer. Then,

Q: Let n be a positive integer and f(n) = -7n50 +2 n1- 40, Then fin) is O None of these O equal to 0(n)…

A: f(n)= O(g(n)) if there exit M>0 and a natural number n0 such that  |f(n)|<M g(n) for all n>…

Q: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

Q: 2-) Find all in tegers x such thet 5x = 12 (mod 7) 10x = 4 (mod 1) 7x E 16 (mod 10d ond 1occ < x <…

A: Given: 5x≡12(mod 7) 10x≡4(mod11) 7x≡16(mod 10) and 1000≤x≤3000

Q: 5. By Principle of Mathematical Induction, prove 2+ 22 + . 2" = 2n*1-2 for all n>1 %3D ...

A: Hi, thank you for the question. Since you have posted two questions, as per our policy we will…

Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

Q: For each statement, indicate whether it is true or false. Then prove or disprove it. (a) For all SC…

A: (a) FALSE.  Consider S ⊆ℤ, defined by  S = { 2+8k | k ∈ℤ } U { 2+12r | r∈ℤ }.  Then the given…

Q: what is (a'+b,a+b) Wher e a and b are relatively prime integers that are not both Zero1

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: Show that if n is an even perfect number, then n = 2P-'Mp for some Mersenne prime Mp 2P – 1.

A:

Q: (6) Compute ind2 (19) modulo 101.

A: We recall the definition of a primitive root.

Q: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

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Q: Find a d, he nce ged (2695,154) other wise express 26 95p +15479=d pigtz the in or

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Q: (c) (n² + n+ 41)² = 1 mod 12 , for every positive integer n

A: We know that,

Q: Show that II (1-) < II p° perfect = n = p*||n p*||n Hint: The following formula should prove useful:…

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Q: Prove 1 0, Ø ! n n=1

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Q: If mand n are positive inte gers with m<n and p is a prime, then pmZpris isomor phic to Z p and F…

A: If m and n are positive integers with m&lt;n and p is a prime, then we have to show that is…

Q: Let aj = 1 and 1 -ап, п >1. n+1 аn+1 Show by mathematical induction that for all positive integers…

A: we have to solve by using mathematical induction

Q: Q6. Let n be n₁ + N₂ + N3 +・・・+nt, where each n; is a positive integer. Use the definition of…

A:

Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

A:

Q: 7. Let p be a prime number and n= 2P – 1. Prove that 2"-1 = 1 (mod n).

A: Let p be a prime number and n=2p-1. Prove that 2n-1≡1 (mod n)

Q: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

A:

Q: If x00 = 1(mod 109) then possible value of x is a) 108 (b) 218 (c) 109 (d) None of these

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Q: Let me Z+ be odd with prime-power factorization m = p₁¹p2²p3³p4, where a1a2 a3 a4 P₁ = P2 = P3 = ±5…

A: As per the question we are given a odd numbers m and its prime power factorization as m =…

Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

Q: Find oll integers Such thot 24! x =1 Cmod 2) 5x = 3. (mod 8) and 1500 < X <2000

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Q: Prove: Let be odd prime. Prove that Aor all i with 1 LikP-2), you have an +... + working so far:…

A: As per the question we just have to show that the prime p divides the following : xi + x2i + ... +…

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A:

Q: 7 .Assune p is an odd prime such that p ≡ 1mod4 (that is p = 4k+1) Prove: if a is a primitive root…

A: Given: p≡1mod4       ∵p=4k+1

Q: Let a € Z be odd. (a) Prove that a = 1 (mod 4) or a = 3 (mod 4). (b) Prove that a² = 1 (mod 4).

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Q: 15. Use a 30-60-90 triangle to find the tangent of 60°. 2 /60° 1 30% √3

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Q: 2. Find the largest negative x € Z that simultancously satisfies 4x = 9 (mod 5) 2x = 0 (mod 6) x =0…

A: Given : 4x = 9 (mod 5) 2x = 0 (mod 6) x = 0 (mod 4)

Q: 20.What is the value of n such that (24  is congruent to 58 mod n) ? a.13 b.17 c.-34 d.-13

A: # as per the guidelines we are entitled to solve one question at a time, please resubmit the other…

Q: 15. Which value can you use for n to show that the first step in a proof by mathematical Induction…

A: Since we can see that option a, b, c are not integers. So they can not be possible because the…

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: 5 how that Cm d n+2 Che prine 4C(n-1)! +)+n = 0 mod @ncnt2)

A: We show that the n and n+2 are primes If and only if  4((n-1)!+1)+n≡0 mod n(n+2) We take n as prime…

Q: e) Let p and q >7 be primes such that 2 +2P +1 =0 mod q. Prove that q = 1+6kp for ome natural number…

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Q: Let m = p^3 qr^3 and n = p^2 r^5, where p,q,r are distinct primes. Find each of the following. a.…

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Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

A: