ythagorean prime number is a prime number of the form 4n + 1, where n ≥ 1. A Mersenne prime nber is a prime number of the form 2" – 1. A perfect number is a number that is equal to the sum of of its divisors, including 1 but excluding the number itself. sider the following statements: Let m is the smallest Pythagorean prime number. Then, 2 – 1 is a Mersenne prime number. If m is a perfect number, then 2™ – 1 is a Mersenne prime number. ect one of the following choices: (1) is True and (2) is False. (1) and (2) are True. (b) (1) is False and (2) is True. (d) (1) and (2) are False.

Q: Proposition 81 For each integer n, n2 - 3n +9 is odd.

A: For each integer given function will be always odd

Q: 1. Let ne Z. Prove that n² +3n - 5 is an odd integer.

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Q: 1. Induction and prime factorisation of natural numbers: Let n EN be a natural number. A prime…

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Q: Prove the following statement by mathematical induction. For every integer n ≥ 0, 7 - 1 is divisible…

A: We need to prove that, for every integer ,  is divisible by .We will prove this using principle of…

Q: Prove the following statement by mathematical induction. For every integer n ≥ 0, 7" - 1 is…

A: We need to prove that, for every integer ,  is divisible by .We will prove this using principle of…

Q: For any integer ? show that the pair of integers 7? + 9 and 4? + 5 are relatively prime.

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Q: nd 24. The reciprocal of any irrational number is irrational. The reciprocal of a nonzero real…

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Q: Prove that if 7 divides n with remainder 1, 7 divides n² + 2n + 7 with remainder 3.

A: Given, if 7 divides n with remainder 1. We have to prove that 7 divides n2+2n+7 with remainder 3.

Q: 2. Make a conjecture about the sum of primes based on the following pattern: 4= 2+2 6=3+3 8=3+5…

A: Given:4=2+26=3+38=3+510=3+712=5+7and so on...

Q: Which of the following correctly define what a prime number is? An integer p > 1 such that if d is…

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Q: Fill in the blanks in the proof of the of fact that 3" – 1 is a multiple of 2 for every natural n.…

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Q: For every integer n, n is even if and only if 3n - 11 is odd. (a) Notice the Result is a…

A: The given biconditional statement is, " For every integer n, n is even if and only is 3n-11 is odd".

Q: c.) There is a conjecture called "Goldbach's weak conjecture" (yes, really!) that every odd number…

A: Given: Odd numbers greater than 5. To prove: Every odd number greater than 5 can be written sum of…

Q: 10. Find all primes p such that 29p+1 is a square.

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Q: 215 = 32, 768. How many numbers between 0 and 215 are the sum of exactly three powers of 8? Each…

A: We need to find the how many numbers between 0 and 215=(23)5=85 such that the sum of exactly three…

Q: Consider the following statement: For all integers a and b, if a is even and b is a multiple of 6,…

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Q: A taxi company has 10 taxis. Part A On a certain day, only seven taxi drivers show up for work. How…

A: Solution: It is given that the taxi company has 10 taxis.

Q: Is the following proposition true or false? Justify your conclusion with a proof or a…

A: Check the following proposition, "For each natural number n, if 3 does not divide (n2+2) then n is…

Q: Let p be a prime number such that p> 2. Use the proof by contradiction to show that at least one of…

A: See the solution attentively I think you will understand easily.

Q: 1. Is the following statement true? If yes prove. If no give a counter example. If a and b are any…

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Q: 1. Prove: If n is odd then 5n + 4 is odd.

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Q: If A = {x : x is a natural number },               B = {x : x is an even natural number}  C = {x : x…

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