Determine all prime numbers a, b and c for which theexpression a² +b² + c² - 1 is a perf

Q: 1) Show that the square of an odd integer is odd. 2) Show that the product of two odd integers is…

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Q: 1. Find all possible pairs of integers whose sum is 30 and product is greater than 216. 2. Find…

A: I have solved first two ,for all parts solution repost again

Q: Use a direct proof to show that the product of two odd integers is odd.

A: Introduction: A direct proof is a series of statements, the final of which is the conclusion to be…

Q: Determine if the statement is True or False.   Show a Counterxample if the statement is False. If…

A: Solution... Consider the two consecutive whole numbers  and : The sum of these two numbers is The…

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: For some real numbers m and n, the list 3n2, m2, 2(n + 1)2 consists of three consecutive integers…

A: For some real numbers m and n, the list 3n2 ,  m2  , 2(n + 1)2  consists of three consecutive…

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: The sum of the squares of the first 5 fibonacci terms, check and prove

A: Fibonnaci Sequence is: 1,1,2,3,5,8,13,21,34,55,89,... (That is, each term of the sequence is the sum…

Q: Prove by using direct proof: Prove that the sum of any three consecutive integers is divisible…

A: Solution   Prove that the sum of any three consecutive integers is divisible by 3.

Q: Choose what method of proof to use always provide solution.   a) Prove that if the sum of any two…

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Q: Find the 50th term of the triangular numbers {1, 3, 6, ..}. Show how you get your answer here:

A: We have to find the 50th term of the triangular numbers {1, 3, 6, …}.

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: Use the Division Algorithm to prove that (a) The square of any integer is either of the form 3k or…

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Q: Use the fact that 12 oz is approximately 6.89pi in3 to find the dimensions of the 12 oz Coke can…

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Q: 10. Determine three positive numbers whose sum is 48 and such that their product is as large as…

A: find the three positive number.

Q: 11. Prove that if n is a perfect square, then n + 2 is not a perfect square.

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Q: 8. We will call the integer n doubly square if n is a perfect square and its number of positive…

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Q: Determine all prime numbers a, b, and c for which the expression  a^2 + b^2 +c^2 -1 is a perfect…

A: To determine all prime numbers a , b and  c  for which expression   a2+b2+c2-1   is a perfect…

Q: Determine all prime numbers a, b and c for which the expression a² + b² + c² – 1 is a perfect…

A: We have to determine all the prime numbers a, b and c for which the expression a2+b2+c2−1 is a…

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

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Q: 18. (a) Show that of all the rectangles with a given area, the one with smallest perimeter is a…

A: Here is the solution

Q: Use direct proof to show that product of two consecutive integers is even.

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Q: Determine all prime numbers a,b and c for which the expression a^2+b^2+C^2-1 is a perfect square

A: Result: Modulo 4 all squares are 0 and 1. 02≡0mod412≡1mod422≡0mod432≡1mod442≡0mod452≡1mod4...

Q: Determine all prime numbers a, b and c for which the expression a² + b? + c² – 1 is a perfect…

A: Result: Modulo 4 of square of integer are 0 or 1.…

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

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

Q: Which of the following numbers is a perfect square or a perfect cube, but not both? 12 544 531 441 O…

A: considering option wise1) 1254412544 =112 , perfect square2) 531441531441=729 , perfect square

Q: Prove that for all primes p2 5, p + 1 cannot be a perfect square. Recall that is a perfect square if…

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Q: Abby notices that for the first twenty perfect​ squares, each square is either a multiple of​ 5, one…

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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: 2. Prove the statement: The sum of the squares of any two consecutive integers is odd.

A: I have assumed that n be any integer. Then i have found the sum of square of two consecutive…

Q: All perfect squares are divisible by 2. A.) True B.) False, counterexample of 16 C.) False,…

A: Numbers that are divisible by 2 are {0,2,4,6,8 ,10 } Perfect sq 1X1=1 2X2=4 3X3=9  4X4=16 5X5=15…

Q: 9. On a real number line, label the points with coordinates of 3 0,1,–1,-,-2.5,- and 0.25 4 2

A: The graph is shown in next step.

Q: Denote by an the nth positive integer that is not a perfect square. For example, we have…

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Q: Find the largest valuc of integer value ofk such that 1 (24 - 1) < 900. 5+7++

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Q: Determine all prime numbers a, b and c for which the expression a² + b° + c° - 1 is a perfect…

A: We have to find the prime numbers a, b and c such that the expression a2+b2+c2-1 is a perfect…

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: We say two numbers m and n are consecutive perfect squares when m is the square of a positive…

A: We need to find year when men was born.

Q: Two consecutive natural numbers are given such that the greater of those number is a perfect square.…

A: Here I am attaching image so that you understand each and every step.

Q: There are three natural number that are less than one thousand that are both perfect square and…

A: Consider the first ten cube numbers 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000. Check how many…

Q: Give a direct proof that if m and n are even are both perfect squares, then mn is also a perfect…

A: Given that, m and n are even and both perfect squares.

Q: difference

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Q: Use the Division Algorithm to prove that (a) The square of any integer is either of the form 3k or…

A: Disclaimer: Since we are not providing hand-written solution right now, the answer you have asked…

Q: Find the sum and product of each of these pairs of numbers (20CBA)16 , (A01)16. Express your answers…

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Q: If m and n are any positive integers and mn is a perfect square, then m and n are perfect squares.…

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Q: Draw all the subgraphs of a square

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Q: Find a polynomich of Jegree 4 with Zeros x = 1₁ x=2, and x=i.

A: Given polynomial has zeros x=1, x=2 and x=i. Therefore (x-1) ,(x-2) and (x-i) is factor of…

Q: Proof by negation   Prove that if an even number is not divisible by four, then it is not a perfect…

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Q: The hypotenuse of a right triangle is 11.0, and one leg is 7.0 units shorter than the other. Find…

A: Given that,the hypotenuse (BC)of a right angle triangle ABCis 11.0 unitsLet, one leg, AB= x unitsand…

Q: the last one is 5 +20

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Q: 9. Prove that the greatest common divisor of two positive integers divides their least common…

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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.