Compute the last digit of 20072009 in two ways: (a) By combining Fermat's Little Theorem with the Chinese Remainder Theorem. (b) By invoking Euler's Theorem.

Q: Determines the numbers (in the corresponding order) of the row in the Pascaltriangle.

A: My student ID is 1908382026. The last two digit's of my ID are 2 and 6. The sum of these two digits…

Q: Determine whether there are infinitely many primes that have a remainder of 12 when divided by 25.…

A: Given question is Determine whether there are infinitely many primes that have a remainder of 12…

Q: .Use Fermat's Little Theorem to find 153603mod 19.

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Q: Find the length of the func tion based Vange de fincd belouw: the on 10<x <29 10

A: We have to find the length of the given function in the range 10 ≤ x ≤ 29. y = 110ex+e-9x

Q: Show that the square roots of two successive natural numbers greater than N2 differ by less than…

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Q: 2. Find the root using Regula-Falsi Method a) x* - x - 10 = 0 Ans. x = -1.697 and 1.855

A: Follow the procedure given below.

Q: Find the least positive integer that yields the remainders 1, 3, and 5 when divided by 5, 7, and 9,…

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Q: A positive integer is divisible by 5, but it leaves remainders of 1 and 7 (resp.) when divided by 3…

A: Here we will use Chinese Remainder Theorem to find solution.

Q: A band of 19 pirates has stolen a sack of coins. When they tried to divide the fortune into equal…

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Q: A positive integer is divisible by 6, but it leaves remainders of 4 and 5 (resp.) Q.3. when divided…

A: Solution: Since the positive integer is divisible by 6, so suppose the required positive integer is…

Q: Compute the entry in the second row, first column of AC.

A: Given, A=9-804-45-89C=858-87220-8-1-75

Q: Find the smallest positive integer that leaves a remainder of 1 when divided by 2, a remainder of 2…

A: Let N be such a number N=10a+9:  N+1 = 10a+10 N=9b+8: N+1 = 9b+9 N=8c+7: N+1 = 8c+8 N=7d+6 and so on…

Q: Make a handwritten resume regarding the Chinese Remainder Theorem by including examples (minimum 2)

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Q: Characterize the primes p for which 63 is a quadratic residue.

A: Quadractic residue: Let p  be any odd prime number and gcd(a,p)=1. Then a is said to be a quadratic…

Q: does there exists   ______such that if n is divided by 5 the remainder is 2 and  if   ?

A: We have to check whether there exist n such that if n is divided by 5 then remainder is 2. Let if n…

Q: Compute [5]-l© [12] in Z17.

A: The solution is 16. (Detail is given in the handwritten notes.)

Q: Find the remainder using Fermat’s Little theorem 452 mod 11.

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Q: 4. Use Chinese remainder theorem to compute 435789 (mod 66).

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Q: Prove that there is no triangular number which is one less than a multiple of 11.

A: Introduction: A triangular number is a positive integer that is the average of two integers. For…

Q: 6. Find the remainder of 17036543 on division by 25.

A: Thus, remainder = 2

Q: b. Find the remainder using Fermat's Little theorem 452 mod 11.

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Q: Use the binary exponentiation algorithm to compute the remainder when 385 is divided by 53.

A: Remainder (385 / 53) 385 = 77 * 5 77 / 53 leaves remainder 24 So  24 * 5 = 120  When 120 / 53 leaves…

Q: Prove that there are 1000 consecutive positive integers that are not perfect squares.

A: Here I give an example. That proves the hypothesis.

Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

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Q: When x is divided by 7, the remainder is 5. What is the remainder when 5x is divided by 7? Moreover,…

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Q: A positive integer is divisible by 6, but it leaves remainders of 4 and 5 (resp.) when divided by 5…

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Q: Multiply three consecutive integers together and then add the second integer to that product. Use…

A: Consider three consecutive numbers x, (x+1), (x+2). Take a product of these numbers we get,…

Q: If f is.q field a hom- saplar monic polphomial in tay can be factored as q, product of monie primes…

A: Any non-zero non-unit element as a product of primes up to order and up to units. Now  If a≠0 and…

Q: 3. According to the Division Algorithm, when 8884113 is divided by 547 a. The quotient is b. The…

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Q: =) Find the remainder of 9045142161 on division by 247 using Euler's Theorem. [Hint: 247 = 13 x 19.]

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Q: the 20 1, 2, 3 Factor the number 211 – 1 by Fermat's factorization method. -

A: Fermat’s Factorization method is based on the representation of an odd integer as the difference of…

Q: 6. Devise a Newton algorithm for computing the fifth root of any positive real number.

A: Given : Positive real number  To Find : Algorithm of Newton method  for fifth root  NOTE : According…

Q: Solve by using the Chinese remainder theorem for the congruence X 2 (mod 7) X 4 (mod 5)

A: Given that This can be written as,

Q: Prove: The product of 2 numbers is equal to the product of their least common multiple and their…

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Q: if two even integers are multiflied, then the product is also even

A: Ans. True

Q: 5. Invent a positive integer a that has at least 8 decimal digits and a ones digit of 7. Estimate…

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Q: 3. Use Euler's Theorem (and the Chinese Remainder Theorem) to how that n12 = 1 (mod 72) for all (n,…

A: The Euler's Theorem states that for a positive integer n and an integer a that is relatively prime…

Q: What is a homogeneous linear recursion of degree k? What is an auxiliary equation of this recursion?…

A: a) What is a homogeneous linear recursion of degree k? b) What is an auxiliary equation of this…

Q: It is true that the division algorithm does not hold for Z[x]. Provide an example to show why.

A: It is given that the division algorithm does not hold for Z[x] Example Let us take a subset S of…

Q: "For all integers n, n(n² – 1) is divisible by 6." (You may use quotient-remainder theorem and…

A: Let us consider, for all integers n a = n3 – n a = n (n2 – 1) a = n (n + 1)(n – 1)

Q: Use the quotient remainder theorem and division into cases to show for all integers n, the integer…

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Q: Prove (NOT BY EXAMPLE) that for every nonzero integer, its square is positive.

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Q: Derive the formula of of fourth order Runge kutta method completely.

A: Assume first order differential equation, dydx=fx,y,y0=y0

Q: Use syntheti division t deknnime k= -i in a zero of explaiin. fex)=x*+2x²+3X +2 the functui and

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Q: In Z[√ - 5], show that 21 does not factor uniquely as a product ofirreducibles.

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Q: Suppose that the number of bacteria in a colony triples every hour.a) Set up a recurrence relation…

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Q: Use Fermat’s little theorem to find the least residue of 863 modulo 31.

A: Fermat's little theorem says: ap≡p(mod p)gcd(a,p)=1

Q: Combine Fermat’s Little Theorem with the Chinese Remainder Theorem to calculate the least residue of…

A: We have to calculate the least residue of 31453modulo 35. That is, we have to find an integer r such…

Q: Find the remainders at each of the four steps of the Euclidean algorithm in the process to find th…

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Q: Prove that the difference of squares of any two odd numbers is múltiple of 8. (SOlve this task…

A: To prove- The difference of square of any two odd numbers is a multiple of 8.   Odd numbers are…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.