Verify that 1225 and 41,616 are simultaneously square and triangular numbers. [Hint: Finding an integer n such that n(n +1) in 1225 is equivalent to solving the quadratic equation n2 +n 2450 0.]

Q: Consider nonnegative integer solutions of the equation xi + x2 + x3 + x4 + x5 + æ6 = 30.

A: The non-negative integer equation is x1+x2+x3+x4+x5+x6=30Since ,xi-number varying from 0 to…

Q: Prove by cases that 25x² + 35x is an even integer whenever 5x-7 is an integer. Note here that the…

A: Apply the condition for 5x-7 to be an integer case by case.

Q: Prove the square of any integer has the form 4k or 4k+1 for some integer k

A: Consider the set of all integers , To prove that the square of any integer is of the form 4k or 4k+1…

Q: 58° of 2.

A: Given: A figure with one of the angles as 58°.

Q: Show that the square of any positive Integer is of the form 3m or 3m+1 for some integer m

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Q: Construct a two-column proof to prove the following by using factoring: Given: a³ - 11a²-9a = -99…

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Q: give all the possible roots of x4+2x3-7x2-8x+12

A: Give all the possible roots of x4+2x3-7x2-8x+12

Q: Show that if a positive integer is odd, then it's square is odd.

A: To show that if a positive integer is odd then its square is odd

Q: Use the definition of even to verify that the expression is an even integer. 12m + 4n+10 2-10m

A: An even integer is an integer which is divisible by 2. Let a is an even integer i.e. a is divisible…

Q: Prove that if n is odd, then 2 is not a square in GF(5n). In other words, prove that there is no…

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Q: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

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Q: Prove that there do not exist three consecutive integer values of nn for which 43n+1443n+14 is a…

A: Given 43n+14 We have to Prove that  ∄  three consecutive integer values of n for which 43n+14 is a…

Q: Prove the square of any odd integer is expressible in the form 8n+1 with n ∈Z.

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Q: Use a direct proof to show that every odd integer is a difference of two squares. (For example: 1 =…

A: first we take a odd integer then we write in 2y+1 form then we transfer to difference of two…

Q: (b) Let z = 24695 + 99288 i. Enter a square root of z, in maple syntax, in the box below.

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Q: Show that the cube for any integer is of the form 7k, or 7k ± 1

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Q: In the 1989, Journal of Number Theory paper, N. Tzanakis and B. de Wegner showed that the largest of…

A: We have to factories the equation  x3+3x2+2x-258,474,216=0

Q: 12 Given that, for all values of x, 4x-8x2-12x + 11 (2x + k)Q(x) + where Q(x) is a quadratic…

A: Quadratic expressions

Q: Factor the following perfect square trinor 49x2 + 14x + 1 O 7(x + 1)2 O (7x - 1) O (x + 7)? (7x +…

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

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Q: Prove that 67n+ 17 cannot be a perfect square for any three consecutive integers n.

A: We will use quadratic residue and proof by contradiction method.

Q: Find all integer solutions to 105x+343y=35 Using Magic Box | Elementary Number Theory

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Q: Prove that if n is odd, then 2 is not a square in GF(5n) In other words, prove that there is no…

A: PROOF :

Q: Suppose that a, b are two integers such that 3a? – b3 = 1. Does it follow that a and b are…

A: Given that a, b are two integers such that 3a2-b3=1.

Q: What is the number of nonnegative integer solutions of the equation x1 + x2 + x3 + x4 + x5 = 15,…

A:

Q: b) Hence or otherwise prove that there do not exist three consecutive integer values of n for which…

A: First let us assume that 29n+5 is a perfect square for some consecutive integers n and n+1. Then…

Q: How many integer solutions to x1 + x2 + x3 + x4 = 32 are there for which x1≥6, x2 ≥5, x3 ≥3 and x4…

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Q: Prove that if n is a perfect square, then n +2 is not a perfect square.

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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: Prove that 4 does not divide n2 +2 for any integer n.

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Q: Consider the statement: "there are no integers and such that 20a+4b%3D1." Is this true or false?…

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Q: Prove that 3/3 is irrational.

A: A number which can be written in the ratio , where and are integers are called rational numbers.…

Q: By using mathematical induction prove that the given equation is true for all positive integers. 9n…

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

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

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Q: find all 4th roots of 1,296   0,  -6

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Q: Prove that 3+√3 is irrational.

A: As per guidelines we are supposed to solve the maximum one questions at  a time kindly post another…

Q: How many nonnegative integer solutions of x₁ + x2 + x3 + 4 = 12 are there in which no r; exceeds 4?…

A: The given problem is to find that number of nonnegative integer solutions for the given equation…

Q: Let m and n be positive integers. Show that mZ+nZ= (m, n)Z.

A: Given: Let m, n be two positive integers. To prove: mℤ+nℤ=m,nℤ. Proof: Let I=mℤ,J=mℤ Since…

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

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Q: Find integers p and q so that 17p + 45q = 1. Show your working steps.

A: Here we have to find the integers p and q so that  17p+45q=1

Q: 1. Use Direct Proof to prove that the product of two consecutive integers is divisible by 2.

A: We have to prove that product of two consecutive integers is divisible by 2.

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: 1. Determine the value of n if one of the roots of 2x2 - 16x + n = 0 is 4-7. |

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Q: Show that 2 is a primitive root of 19.

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Q: Prove that the square of any odd integer n, (n =4q +3 for some integer q) has the form 8m + 1 for…

A: Here we have to prove that any odd integer n which is of the form (n = 4q + 3) , where q is an…

Q: 1) What could be added to each expression to make it a perfect square? 1) a +12x 2) z2-6z+1 3) 2…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: Ф(тk) — ф(0) — тk

A:

Q: 3. Prove that if n is a triangular number, then so are 9n +1, 25n + 3, and 49n + 6. Hint: Recall…

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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…