course Mth 151 E¤ð—™R“©¶Íã£]’à†è›Ù‹Ã´assignment #024
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15:03:40 `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 --> The sum of 13 and 15 is 28 confidence assessment: 2
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15:03:54 28 can be expressed as 5 + 23, both of which are prime.
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RESPONSE --> true self critique assessment: 2
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15:11:09 `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 = 17+59 confidence assessment: 2
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15:11:35 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: 2
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15:23:59 `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 --> the prime factors of 18 is 1,2,3,6,9 and this adds up to 21 which is greater than 18 itself confidence assessment: 2
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15:24:18 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 --> that's what i got self critique assessment: 2
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15:30:27 `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 --> 12= 1,2,3,4,6 which = 16 making it abundant 26= 1,2,13 which =16 making it dificient 16 = 1,2,4,8 = 15 making it dificient 6 = 1,2,3 = 6 making it perfect confidence assessment: 2
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15:30:57 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 --> self critique assessment: 0
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15:39:06 `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 no's of 28 is 1,2,4,7,14 which adds up to 28 confidence assessment: 2
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15:39:38 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 --> self critique assessment: 0
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15:40:13 `q006. Why can't a prime number be perfect?
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RESPONSE --> because you don't include the number itself to proper factors confidence assessment: 2
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15:41:32 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 --> self critique assessment: 0
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15:43:51 `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 --> this is not true if I took the number 4 and placed it for n you would get 15 and 15 is not a prime number. confidence assessment: 2
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15:44:10 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 --> self critique assessment: 0
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15:48:33 `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 --> 3 gives you a prime, 5 gives you a prime and 7 gives you a prime confidence assessment: 2
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15:48:48 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 --> self critique assessment: 0
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