Solve the following problems in essay form (1 to 2 paragraphs for each problem is fine). Please show all work and show how you came up with the answer. Please make sure that you address the following questions: (1) How did you get your answer? (2) What steps did you take? (3) Where did you begin? (4) Why did you do what you did? (5) Explain in detail what your answers to the problems are?

Problem 1 -
Mr. Smith invited 17 guests to his "Disco-reunion" party. He assigned each guest a number from 2 to 18, keeping 1 for himself. When everyone was dancing, he noticed that the sum of each couple's numbers was a perfects square. What was he number of Mr. Smith's partner?

Problem 2 -
Define "prime date" to be a date during the year where both the months (as a number, i.e. January is 1, February is 2, etc) and the day are both prime numbers. What is the average number of prime dates per week in one year?

Thank you

Solution Preview

Please see the attached file.

Problem 1 -
Mr. Smith invited 17 guests to his "Disco-reunion" party. He assigned each guest a number from 2 to 18, keeping 1 for himself. When everyone was dancing, he noticed that the sum of each couple's numbers was a perfect square. What was the number of Mr. Smith's partner?
We want to find a number between 2 and 18, which when added to 1 makes a perfect square. This is because Mr. ...

Solution Summary

This is a pair of word problems involving prime numbers and perfect squares.

... If there is a prime factor greater than the square root of that integer, can the integer be a perfect square? The maximum number of prime factors greater ...

... Prove that if n is a perfect number , then ∑1/d = 2. d/n. ... α Let n = p1 1 p 2 2 ..... prr be the prime factorization of ... p rer where pi ,s are odd primes. ...

... column is located between the question numbers and the ... 1 The prime factorization of the integer 30 is ... a difference of squares, a perfect square trinomial, just ...

... 2 and 3 are prime numbers but their sum 2 + 3 = 5 is ... Thus, next three numbers will be 1/81, 1/243, 1/729. ...squares but their sum 4 + 9 = 13 is not an even number. ...

... is the multiplicative group of non-zero complex numbers. ... Friedrich Gauss proved that for any prime p the ... Verify this statement for all primes ≤ 17 giving p ...

... where x is considered to be a real number (imaginary or complex numbers not included ... of the numerator and denominator and deleting those prime factors which ...

... The LCM is the product of all the prime numbers write down ... Is the following set of numbers a perfect triple ... work: to be a perfect triple for a set of number a, b ...

... is 1 *7 = 7, which is a prime number, so the ... 4 - 9n 4 Using the difference of squares ( 3n ) + 3n ... 12xy + 4y 2 - 25 First using the perfect square to factor ...

... Y' reads Y prime, and it is the predicted value of the ... If the number of plus or minus signs is more ... eg converting all differences to positive numbers); sum the ...

... while still requiring that m be prime and that ... aLength]; //get the frequency of random numbers in the ... rand() % 10]; //use your random number generator instead ...