2. Make a conjecture about the sum of primes based on the following pattern:(4=2+26=3+38=3+510=3+712 = 5+7and so on.O The sum of primes is equal to the multiple of the primes.O The sum of primes is between 4 and 12.The sum of primes is equal to an even number.O The sum of primes is equal to primes.

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

Answered: 2. Make a conjecture about the sum of…

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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1.) Factor 51, 948 into a product of primes.
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1. Let p be a prime number (integer). Show that p? + 5 cannot be a prime number.
5. Prove that for any integer n, at least one of the integers n, n+2, n +4 is divisible by 3. Your answer needs to be a little bit longer. Write a few sentences to complete your assignment. 6. A classic unsolved problem in number theory asks if there are infinitely many pairs of "twin primes", pairs of primes separated by 2, such as 3 and 5, 11 and 13, or 71 and 73. Prove that the only prime triple (i.e. three primes, each 2 from the next) is 3, 5, 7. Your answer needs to be a little bit longer. Write a few sentences to complete your assignment. 7. Prove that for any natural number 1, 2+2² +2³+. +2²² = 2n+1 2
Find integers q and r such that 250=16q+r and 0≤r<16 . Explain carefully why gcd(250,16)=gcd(16,r). Check that this equation is in fact true by explicitely finding all the factors of 250, 16, and r. (One possible way to list the factors is to first find the prime factorization of each number.)
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It is believed that every even number is the sum of two prime numbers. Two prime numbers whose sum is 32 are A) 7 ,25 B) 11 , 21 C) 13 , 19 D) 15 , 17
Show that if a and b are relatively prime then (a+2b,2a+b)=1 or 3.
3. Kate was given the following number pattern by her teacher. 22+1=5 42+1=17 62+1=37 102+1=101 14²+1=197 Kate made the following conjecture based solely on the number patterns shown. "When an even number is squared and one is added to it, the resulting number is a prime number". In the examples above 5, 17, 37, 101, and 197 are prime numbers because the only factors of these numbers are 1 and the number itself. a) Is Kate's conjecture True or False, based solely on the numbers shown? Use your understanding of even numbers and prime numbers to explain your answer. b) Determine a counterexample to the pattern above to show that Kate's conjecture is false.
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