7. Suppose a1, a2,...,ɑn are integers, r is a solution to the equation 1" + a,r"-1 + a2r"-² + + a, = 0. ... Prove that if r is rational, then it is an integer.

Q: Prove that Va - Vb is irrational number if 'a' and 'b’ are prime number?

A: Given To prove that a-b is irrational number if a and b are prime number.

Q: How do I prove that if a, b, and c are integers with c not equal to 0 then ac = bc => a = b?

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Q: 2. Prove by mathematical induCHon that १) के) 22 32 42 2n is Hrue For all integers nz 2

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Q: (b) Prove or disprove: Z10 ~ Z5 x Z₂.

A: We shall solve first question only as you have asked more than one unrelated question. For others…

Q: Prove: If x, y, z are integers with x2 + y? = z², then either x or y (or both) is even. Hint: Do a…

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Q: Discrete

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Q: write down the proof of the statement " √2 + √5 is not rational" in latex

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Q: 3/4 Given: 4 2x+ 35 Prove: x= 4

A: Given: 42x+34=35

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

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Q: Let a and b be positive integers. If ax + by = c has an integer solution (x, y), it is necessarily…

A: Consider the given equation. Find the characteristic for c if a, b are positive integers and ( x, y…

Q: (2) Prove by contradiction that if 3z2- is irrational then r is irrational. 215

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Q: 6. Evaluate In (x - In (r-...))dr in terms of some new variable t (do not simplify).

A: Given :- ∫lnx-lnx-...dx To find the given integral in terms of some new variables t

Q: Show that 3n^ 2+8nlogn is big-Oh (n ^2 )

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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)…

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Q: Prove that in Q(/-5) we have (3,1+ 2v-5)(3, 1 – 2-5) = (3).

A: To show the equality, first we take an element of the product and show that it is in (3) and then we…

Q: 1. If today is Saturday, what day of the week will it be 1,500 days from today? 2. Prove that 4/2 +…

A: The explanation of the solution as follows:

Q: 9. Prove that V2 is an irrational number.

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Q: Prove by contradiction: If x is a nonzero rational number and y is an irrational

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Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: 3. Prove by contradiction, that if x and y are odd integers, then there is no integers z such that…

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Q: 3) Prove that there do not exist two rational numbers whose sum is irrational.

A: To prove: There do not exist any two rational numbers such that their sum is irrational.

Q: 2. Prove that V6 is irrational number.

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Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

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Q: s) Express the infinite contimod fraction [2, 1,5 quadratic irrational- as a ity.

A: We need to express the infinite continued fraction 2, 1, 5¯ as a quadratic irrationality. The…

Q: prove that 3^1/2 is irrational

A: We have to prove that 3^1/2 is an irrational number.Let us assume that 3^1/2 is a rational number.…

Q: is the completion of CX sd) is the set of all rational number

A: We know that a metric space (X,d) is complete iff every Cauchy sequence in X is convergent in X but…

Q: 15. Find the general solution of dr bcosedr + rsinode

A: General solution of differential equation

Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

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Q: Prove that (3+2/5) is irrational

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Q: Show that there is no rational number r satisfying 2" = 6. Where does your proof break down if we…

A: Let us consider that, suppose a rational number r exists such that r=pq; q≠0. Now,  2pq=6 On taking…

Q: 6. Prove that if x is an irrational number and Y is a nonzero rational number, then xy is…

A: to prove this we take xy is rational number then show this is contradiction.

Q: Show how to adapt the proof that v3 is irrational (from the video) to prove that v15 is irrational.…

A: 3 is irrigational number: Let 3 is rational number then 3=pq,where p, q are both integers and q≠0…

Q: 3. The v77 is between what two integers?

A:

Q: Prove that 21 divides 2n-) 4nt whenever is Posi tive in teger in

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Q: b) Prove or disprove: Z10 ~ Z5 x Z₂.

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Q: Prove, that (nezond n73) If x"=5,+hen x is not a rational number.

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Q: b. Prove that 2 is irrational.

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Q: (b) Prove that 2m ≤ m! + 2 for m ¤ N. (c) For any nonzero real number x, prove ---3 i=1 = 1- x -n

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Q: Suppose that a, b. are two integers such that 3a² – b = 1. Does it follow that a and b are…

A: Given that,

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: 7. Suppose a1, a2,...,an are integers, r is a solution to the equation x" + a1x"- + a2x"- +.. + an =…

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Q: List the first 3 positive prime numbers d in Z so that the quadratic integers in Q[Vd] are precisely…

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Q: show that there does ot exist a rational numbers such that s2 = 6

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Q: 1. Prove that 3 is irrational number.

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Q: 9. Prove that V3 and /2 are irrational by assuming that each is rational and arguing until a…

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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?…

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Q: prove that 2-2.7 + 2 + マ(3) = 2(-7)" = %3D (1-(-ス)^+) 141 wheneer is Cre nonnega h'Ve integer

A: we will prove it by mathematical induction