#$&* course mth151 12/17/14 9:20 If your solution to stated problem does not match the given solution, you should self-critique per instructions
.............................................
Given Solution: 28 can be expressed as 5 + 23, both of which are prime. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: `q002. The assertion that any even number greater than 4 can be expressed as a sum of two primes is called Goldbach's conjecture. Verify Goldbach's conjecture for the numbers 42 and 76. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 42=23+19 76=23+53 confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: 42 = 23 + 19, or 13 + 29, or 11 + 31, or 5 + 37. 76 = 73 + 3, 71 + 5, 59 + 17, 53 + 23, or 29 + 47. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: `q003. The proper factors of a number are the factors of that number of which are less than the number itself. For example proper factors of 12 are 1, 2, 3, 4 and 6. List the proper factors of 18 and determine whether the sum of those proper factors is greater than, less than, or equal to 18 itself. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 18 1,2,3,6,9 1+2+3+6+9=21 21 is greater than 18 confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: The proper factors of 18 are easily found to be 1, 2, 3, 6 and 9. When these factors are added we obtain 1 + 2 + 3 + 6 + 9 = 21. This result is greater than the original number 18. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: `q004. A number is set to be abundant if the sum of its proper factors is greater than the number. If the sum of the proper factors is less than the number than the number is said to be deficient. If the number is equal to the sum of its proper factors, the number is said to be perfect. Determine whether each of the following is abundant, deficient or perfect: 12; 26; 16; 6. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 12 1,2,3,4,6 1+2+3+4+6=16 12 is abundant 26 1,2,13 1+2+13=16 26 is deficient 16 1,2,4,8 1+2+4+8=15 16 is deficient 6 1,2,3 1+2+3=6 6 is perfect confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: The proper factors of 12 are 1, 2, 3, 4 and 6. These proper factors add up to 16, which is greater than 12. Therefore 12 is said to be abundant. The proper factors of 26 are 1, 2, and 13. These proper factors add up to 16, which is less than 26. Therefore 26 is said to be deficient. The proper factors of 16 are 1, 2, 4 and 8. These proper factors add up to 15, which is less than 16. Therefore 16 is said to be deficient. The proper factors of 6 are 1, 2, and 3. These proper factors add up to 6, which is equal to the original 6. Therefore 6 is said to be perfect. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: `q005. There is a perfect number between 20 and 30. Find it. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 21 1,3,7 deficient 22 1,2,11 deficient 23 prime so deficient 24 1,2,3,4,6,8,12 abundant 25 1,5 deficient 26 already seen this was deficient 27 1,3,9 deficient 28 1,2,4,7,14 1+2+4+7+14=28 perfect confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: The numbers 23 and 29 are prime, and no prime number can be perfect (think about this for a minute and be sure you understand why). 20 has proper factors 1, 2, 4, 5 and 10, which add up to 22, so 20 is abundant and not perfect. 21 has proper factors 1, 3 and 7, which add up to 11, which make 21 deficient. 22 has proper factors 1, 2 and 11, which add up to 14, so 22 is deficient. 24 has proper factors 1, 2, 3, 4, 6, 8 and 12, which add up to 35, so 24 is abundant. 25 has proper factors 1 and 5, and is clearly deficient. 26 was seen earlier to be deficient. 27 has proper factors 1, 3 and 9, and is clearly deficient. 28 has proper factors 1, 2, 4, 7 and 14. These add up to 28. So 28 is a the perfect number we are looking for. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: `q006. Why can't a prime number be perfect? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: a prime only has two factors 1 and itself that means it would only have a proper factor of 1 and all prime numbers are greater than 1 so they cant be perfect confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: A prime number has only two factors, itself and 1. It therefore has only one proper factor, which is 1. Since every prime number is greater than 1,no prime number can be perfect. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: `q007. 2^2 - 1 = 3, which is prime. 2^3 - 1 = 7, which is prime. 2^5 - 1 = 31, which is prime. Is it true that for any n > 1, 2^n - 1 is prime? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2^4-1 16-1=15 15 is divisible by 3 and 5 so it is not prime confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: You might at first think that the above examples establish a never-ending pattern, but if you check n = 4 you find that 2^n - 1 = 2^4 - 1 = 16 - 1 = 15, which is divisible by 3 and 5 and is not prime. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: `q007. Check for p = 3, then p = 5, then p = 7 to see whether the formula 2^p - 1 seems to give primes, where p stands for a prime number. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2^3-1 8-7=7 2^5-1 31-1=31 267-1 128-1=127 all of these all prime but it might not always work out that way confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: 2^3 - 1 = 8 - 1 = 7. 2^5 - 1 = 32 - 1 = 31. 2^7 - 1 = 128 - 1 = 127. All these results are prime. However this doesn't prove that the formula always works. Your book will address this question. ********************************************* Question: `q008. What is the value of 2^p - 1 for p = 11? You aren't asked here to verify whether the number you have calculated is prime. However verify in the most efficient way you can, without the use of a calculator, whether 2, 3, 5 or 7 are divisors of your number. Assuming that your candidate number is in fact prime, list the remaining numbers by which you would have to divide in order to verify this. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2^11-1 2048-1=2047 not divisible by 2 because it isnt even 2+0+4+7=13 not divisible by 3 because the sum of the digits is not divisible by 3 not divisibile by 4 because it isnt divisible by 2 not divisible by 5 because it doesnt end in 0 or 5 not divisible by 6 because its not divisible by 2 and 3 we just have to work out on paper if its divisible by 7. we get 2047/7=292 with a remainder of 3 not divisible by 8 since the last 3 digits dont form a number divisible by 8 not divisible by 9 since the sum of the digits is not divisible by 9 not divisible by 10 since it doesnt end in 0 not divisible by 12 since its not divisible by 3 and 4 since we still have not found a number that 2047 is divisible by we can start dividing the prime numbers leading up to the square roots of 2047, which is a little over 45 we know 2,3,5,7 dont work so the next prime numbers to try are 11,13,17,19,23,29,31,37,41,43 11,13,17,19 do not work but once we get to 23 we find that 2047/23=89 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-critique Rating:ok " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!