PLEASE HELP WITH ALL QUESTIONS. THANK YOU 4. 5. In constructing the Sieve of Era

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PLEASE HELP WITH ALL QUESTIONS. THANK YOU

4.

5.

In constructing the Sieve of Eratosthenes for 2 thrcugh 100, we said that any composite in that range had to be a multiple of some prime less than or equal to 7 since the next prime, 11, is greater than the square roct of 100. Answer the following questions. To extend the Sieve of Eratosthenes to 200 what is the largest prime whose multiples would have to be considered? Complete this statement: In seeking prime factors of a given number, we need only consider all primes up to and including the least one other prime facor less than the Complete this statement: If no prime less than or equal to n divides n, then n is a | of that number, since a prime factor greater than the can only occur if there is at number

Explanation / Answer

1. To extend the Sieve of Erastothenes to 200, the largest prime would be 14.

152 = 225 > 200.

If we consider the prime factors of a number, we need to only consider primes upto the square root of the number since a prime factor greater than the square root can only occur if there is another prime factor lesser than the square root.

If no prime less than or equal to n divides n, then n is a prime number.

2. 5,3,17,29,37,41.

The answer is C. 37 = 12 + 62 41 = 42 + 52.

3. Since 26 (27 - 1) is of the form 2p-1 (2p - 1), it is a perfect number.

Answer is true.

4. The four abundant numbers are 12, 18, 20, 24.

5. 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315

= 975

The correct answer is:

A. The sum of proper divisors is 975 which is greater than 945.

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