The sum of the squares of two odd integers is 1) Perfect 2) Prime 3) Not perfect

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

A: Types of integers : Odd integers : The integer which is not divided by 2 . It has the form 2k+1…

Q: The diagram below shows ten circles joined by five lines with three circles Place all the…

A: Requirements: Numbers should range from 1 to 10.   Every line has three circles.   3. The amount for…

Q: Two distinct numbers are selected from the set (1,2,3,4,...,36,37) so that the sum of the remaining…

A: Given two distinct numbers are selected from the set 1,2,3,4,…,36,37 and The sum of remaining 35…

Q: 4. An open box is to be made from a square piece of cardboard, 18 inches by 18 inches, by cutting…

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Q: How many four digit positive integers are there that contain the digit 3 and are divisible by 5?

A: A number is divisible by 5, if the unit digit place contains the number 0 or 5. Let m choices to be…

Q: How many five-digit numbers are there for which each digit is either equal to both adjacent ones, or…

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Q: 4. Show that if k is a triangular number, then 8k + 1 is a perfect square.

A: Since you have posted a multiple question according to guildlines I will solve first question for…

Q: Find the smallest distinct postive numbers that provide a counterexample to show the statement is…

A: Let the odd numbers be 1 and 3 and the even number be 2, According to the question, The sum of any…

Q: Consider all positive integers that are mueiples of 20 and that are less than or equal to 300. What…

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Q: 8. How many positive integers less than 1000 a) have exactly three decimal digits? b) have an odd…

A: According to our guideliness we can answer only the first 3 subparts. Positive integers less than…

Q: (a) Show that if seven integers are selected from the first 10 positive integers, there must be at…

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Q: A building has 13 smaller units of identical dimensions of rectangular shape and 5 larger units…

A: Given that,1. A building has 13 identical rectangular units and 5 identical square units. 2. Square…

Q: When 5 is added to natural number n it gives a perfect square and when 11 is subtracted from n it…

A: When 5 is added to natural number n it gives a perfect square and 11 is subtracted from n it also…

Q: What is the largest number in the row starting "1 4 ..." of Pascal's triangle?   What is the sum…

A: To find what is the largest number in the row starting "1 4 ..." of Pascal's triangle? To find what…

Q: Prove that the sum of any five consecutive positive integers is always a multiple of five.

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Q: How many pairs of two distinct integers chosen from the set {1, 2, 3,..., 101} have a sum that is…

A: Since you have posted multiple questions and we can answer one question. So answer provided for the…

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: How many pairs of integers (a,b) exists such that the product of a,b and HCF of (a,b) is 1080??

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Q: How many different integers must be selected to guarantee that at least 5 of them would have the…

A: Dividing any integer by 3, can have remainder 0, 1 or 2. Therefore, possibilities for remainder are…

Q: Find a three-digit positive integer such that the sum of all three digits is 14, the tens digit is…

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Q: Prove by contradiction that, if n is a positive integer which is not a perfect square, the sqrt(n)…

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Q: Show that there are infinitely many prime numbers.  0 /∈ N. (zero is not an element of natural…

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Q: Show that given any set of 5 natural numbers, there are 3 numbers whose sum is divisible by 3.

A: Let q∈N. Note that, for every q there exists some x∈N and some v∈0,1,2 such that q=3x+v. Now, let…

Q: 2. Find all Pythagorcan Triples (not just primitive ones), containing the integer 15.

A: The Pythagorean triples are positive integers a,b,c such that a2+b2=c2. We can say that the…

Q: Number theory.

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Q: Prove that if the sum of the digits is subtracted from a natural number, then the result is…

A: let us assume the number to be three digit number xyz therefore, xyz=100x+10y+z sum of digits will…

Q: Construct four relatively prime integers a, b, c, d such that no three of them are relatively prime.

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Q: How many pairs of two distinct integers chosen from the set {1, 2, 3, 101} have a sum that is even?

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Q: Prove in a thoroughly written proof that for all integers a  and m, if a and m are the lengths of…

A: Recall the following. Pythagoras theorem: For any right triangle, the square of the hypotenuse is…

Q: Prove that the sum of the squares of three consecutive integers cannotbe a square.

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Q: What two positive real numbers whose product is 50 have the smallest possible sum?

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Q: Prove that no number of form 4k+3 (where K is a positive integer) can ever be the sum of two perfect…

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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: What is the smallest sum possible sum of 25 consecutive positive integers which s a perfect square?

A: Let's take the 25 consecutive positive numbers as an Arithmetic Progression with first term a.…

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: What is the set of O of odd positive integers less than 10?

A: The odd positive integers less than 10 are 1, 3, 5, 7, 9.

Q: 1. Show that if seven integers are selected from the first 10 positive integers, there must be at…

A: We can solve the following question by pigeonhole principle

Q: Show that if seven integers are selected from the first 10 positive integers, there must be at least…

A: We can solve this question by pigeonhole principle

Q: that the set of negative integers divisible by 3 is countable

A: Set of negetivee integers divisible by 3 isS=-3n, n∈ℕf: ℕ→S   defined asf(n) = -3n,To check…

Q: What is the contrapositive of the statement “ A rectangle R with side lengths 1 and x, where x is…

A: theory Indeed, the ratio between length and width must be rational, even if we don't require the…

Q: two diagonally opposite corners of a standard chessboard are removed. Is it possible to cover the…

A: Given that, Two diagonally opposite corners of a standard chessboard are removed. Every domino in a…

Q: Show that a 6 × n board (n ≥ 2) can be tiled with L-shaped tiles, without gap and overlapping. Each…

A: Before we begin proving the statement, we shall understand the shapes. An L-shaped three-square tile…

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

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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…