Use the quotient remainder theorem and division into cases to show for all integers n, the integer n2 has a remainder of 0 or 1 when divided by 3.

Q: 91. Use the Factor Theorem to prove that x - c is a factor of x" – c" for any positive integer n.

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Q: show that the following number is rational by writing it as a ratio of two integers 5.6077

A: Rewrite the given number as: 5.6077 = 5+0.6077            = 5+607710000 only factors of 10000 are 5…

Q: Show that every rational number has exactly two finite simple continued fraction expansions.

A: To Show : Show that every rational number has exactly two finite simple continued fraction…

Q: Perform the following division problem. Make sure you show pictures of the base 10 or other base…

A: Given: 301 ÷ 4  in base 5 As the base is 5, collect the blocks of unit 5 and blocks of unit 1 to…

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: show that the following number is rational by writing it as a ratio of two integers. 2 9  +  2 3

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Q: In a group, there were 115 people whose proofs of identity we passport, some had voter id and some…

A: “Since you have posted a question with multiple sub-parts, we will solve the first three subparts…

Q: Let N = 7^11. Find the remainder of dividing 8^N by 13.

A: We have N=711 Since 711=1977326743 So we can write 8N=81977326743 We can write it as…

Q: Show that the following number is rational by writing it as a ratio of two integers. 8/9 +2/5

A: A rational number is a real number that is the ratio of two integers. There are certain criteria for…

Q: Prove the formula for the sum of the first positive integers:

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Q: Find the canonical prime factorization of the following using Fermat’s Method and solve for the…

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Q: Use ibn al-Haytham's procedure to derive the formula for the sum of the fifth powers of the…

A: We have to derive the formula 15+25+......+n5=n66+n52+5n412−n212 by using ibn al-Haytham's method…

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

A: Thus, remainder = 2

Q: Show that if 5 divides n2 with remainder 4, 5 divides 3n3 with remainder 2.

A: Given that 5 divides n2 5 n25 n×n5 n5 n3 As it gives the remainder 4 on dividing by 5. Therefore it…

Q: Show that n22-1 is division by 8, if n is an odd positive integer

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

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Q: Show that the following number is rational by writing it as a ratio of two integers. 0.74747474...

A: l

Q: 4. Prove that the product of any two consecutive integers has the form 3k or 3k+2 for some integer…

A: we are entitled to solve only 1 question at a time so I am providing you the same. Kindly repost the…

Q: Find the sum of the positive even integers up to and including 50.

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Q: Find the remainder of 123456789 is divided by 11. Show complete solutions.

A: We will find the remainder.

Q: show that the following number is rational by writing it as a ratio of two integers 0.74747474…

A: According to the given information, it is required to write the given number as a rational number.

Q: using the quotient remainder theorem, show that any integer n can be written in one of the four…

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Q: Let a = -26, 6 = 12 For a divided by b, what is the integer quotient using the least nonnegative…

A: To find: The integer quotient and remainder using the least nonnegative remainder method and least…

Q: Prove by contradiction that there is no greatest multiple of 5.

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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: Find the canonical prime factorization of the following using Fermat's Method and solve for the…

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Q: Use the method of finite differences to find a closed polynomial formula for the following sequence…

A: 1, 3, 21, 91, 273, 651, 1333, 2451, 4161, 6643,... find the first difference D1=2,18, 70, 182,…

Q: Find the smallest positive interger x such that 1001 x leaves a remainder of 70 when divided by 315

A: The given statement can be expressed as an expression, which ix given below  315y+70=1001x Now,…

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

A: Ans. True

Q: Prove that for any integer a, 9/(a – 3).

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Q: Find the largest negative integer leaves a remainder 2 when divided by 3, 5, and 12. (If it is not…

A: Let us  solve the question in the next step.

Q: 6. Find the remainder when 721 divided by 43 7. For a given integer n, prove that 3 k-0

A: To find the remainder when 7221 divided by 43. Now, 7 and 43 are relatively prime and 43 is a prime…

Q: Use the recurrence 1 )=(5x³ – 3x) - A P(x), as a series of Legendre polynomials. given formula to…

A: Given: P3(x)=125x3-3x and P2x=123x2-1 To show: P5(x)=1863x5-70x3+15x using recurrence formula and…

Q: The binomial theorem states that for any real numbers a and b, (a +b)" = E o (") a"-* for any…

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Q: Using modular arithmetic and Fermats Little Theorem, what is the remainder of (x-1)! when divided by…

A: Let   x  be prime. We need to find the remainder of  x - 1!  when divided by x. Since we know:   x -…

Q: 1. Suppose that you take an integer, n, apply the Quotient-Remainder Theorem where d = 7, and find…

A: The quotient-remainder theorem is expressed as n=dq +r and 0≤r<d where qand r are quotient and…

Q: Use the Division Algorithm to prove that for all positive integers, n,6|(n(n + 1)(2n + 1)).

A: We need to prove that, for all positive integers n, 6 is a factor of the expression given below.

Q: Use the method of proof by cases to prove that for any integer n the product n(n + 1) is even

A: Given,  for any integer n the product n(n + 1) is even  We have to prove it using the method of…

Q: 8. Show that no prime number of the form 4k +3 can divide a number of the form n2 + 1.

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Q: 5 Let flx) -5X+3Xh emd gx) = 3X+2X+) in 7[x].Determine the quotient ane remainder upon dividing #00…

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Q: Prove that C(n + 4, r + 1) = C(n + 3, r + 1) + C(n + 3, r). Hint: expand each Combination and use…

A: Explanation of the answer is as follows

Q: Please explain and provide a solution: Remainder 0 or 1 is given when n2 is divided by 3 ( n for all…

A: According to Quotient remainder theorem, for any non zero integers n and d, there exist integer q…

Q: Suppose n is odd. Find the cube of n and divide it by 4. What possible remainders could occur? Check…

A: Solution

Q: Prove by induction that the product of the first n even numbers is equal to n!2" for all positive…

A: We have to prove by induction that the product of first n even numbersis equal to n!2n for all…

Q: Use induction to prove: for any integer n > 1,4 evenly divides 9"

A: for any integer n≥1, 4 evenly divides 9n-1

Q: Prove that the integer (1!)(2!)(3!)· · ·(398!)(399!)(400!)/200! is a perfect square. (In this…

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Q: Use the Division Algorithm to show that the cube of any integer is of the form 9k, 9k + 1 or 9k + 8.

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Q: Show that 19 divides 10*8n-1 + 33n-1 for every positive integer n.

A: Concept: Mathematical induction: For each and every natural number n, mathematical induction is a…

Q: Use the Chinese Remainder Theorem to find a solution to the congruences х 2 тоd 10 x 3 mod 1001 х 3…

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