course Mth 151 ???|???????assignment #024024. More number theory
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13:11:31 `q001. There are seven questions in this assignment. Pick any even number--say, 28. It is believed that whatever even number you pick, as long as it is at least 6, you can express it as the sum of two odd prime numbers. For example, 28 = 11 + 17. Express 28 as a some of two prime factors in a different way.
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RESPONSE --> 28 can also be expressed as: 23+5, which are both prime numbers confidence assessment: 3
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13:11:37 28 can be expressed as 5 + 23, both of which are prime.
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RESPONSE --> OK self critique assessment: 3
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13:12:19 `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.
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RESPONSE --> 42 = 19 +23 76 = 59 +17 Golbach's conjecture is true for these two examples. confidence assessment: 3
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13:12:23 42 = 23 + 19, or 13 + 29, or 11 + 31, or 5 + 37. 76 = 73 + 3, 71 + 5, 59 + 17, 53 + 23, or 29 + 47.
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RESPONSE --> OK self critique assessment: 3
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13:13:46 `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.
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RESPONSE --> Proper factors of 18: 1, 2, 3, 6, 9 1+2+3+6+9 = 21 21 is greater than 18 18 is abundant because it is less than the sum of it's divisors confidence assessment: 3
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13:13:51 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.
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RESPONSE --> OK self critique assessment: 3
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13:16:11 `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.
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RESPONSE --> Proper factors of 12: 1, 2, 3, 4, 6 1+2+3+4+6 = 16 12 is less than the sum of its divisors, so it is abundant Proper factors of 26: 1, 2, 13 1+2+13 = 16 26 is deficient because it is greater than the sum of its divisors Proper factors of 16: 1, 2, 4, 8 1+2+4+8 = 15 16 is deficient because it is greater than the sum of its divisors Proper factors of 6: 1, 2, 3 1+2+3 = 6 6 is a perfect number confidence assessment: 3
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13:16:22 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.
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RESPONSE --> OK self critique assessment: 3
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13:17:37 `q005. There is a perfect number between 20 and 30. Find it.
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RESPONSE --> The perfect number between 20 and 30 is 28. The proper factors of 28 are: 1,2,4,7,14 1+2+4+7+14 = 28 confidence assessment: 3
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13:17:41 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.
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RESPONSE --> OK self critique assessment: 3
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13:18:33 `q006. Why can't a prime number be perfect?
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RESPONSE --> A prime number's factors are only 1 and itself. A prime number's proper factor is only 1. With nothing to add to 1 (because there are no other proper factors), no prime number can be perfect. confidence assessment: 3
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13:18:39 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.
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RESPONSE --> OK self critique assessment: 3
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13:19:47 `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?
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RESPONSE --> The questions uses 2,3,and5 to test the formula. Using 4: 2^4 - 1= 16-1 = 15 15 is not a prime number, so the formula is not always true confidence assessment: 3
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13:19:52 You might at first think that the above examples establish a 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.
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RESPONSE --> OK self critique assessment: 3
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13:21:02 `q007. Check for p = 3, then p = 5, then p = 7 to see whether the formula 2^p - 1 seems to give primes.
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RESPONSE --> p=3 2^3 - 1 = 8-1 = 7 p=5 2^5 -1 = 32-1 = 31 p=7 2^7 -1 = 128 -1 = 127 These are all prime. confidence assessment: 3
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13:21:06 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.
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RESPONSE --> OK self critique assessment: 3
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