#$&* course Mth 151 023. `query 23
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Given Solution: `a** If the number isn't too big we can simply start dividing by primes, beginning with the smallest: If we divide 172 by 2 we get 86, so 2 and 86 are factors. If we divide 172 by 3 we get 57 with a remainder so 3 isn't a factor. If we divide 172 by 4 we get 43, so 4 and 43 are factors. If we divide 172 by 5 we get 34 with a remainder so 5 isn't a factor. If we divide 172 by 6 we get 28 with remainder so 6 isn't a factor. If we divide 172 by 7 we get 24 with a remainder so 7 isn't a factor. If we divide 172 by 8 we get 21 with remainder so 8 isn't a factor. If we divide 172 by 9 we get 19 with a remainder so 9 isn't a factor. If we divide 172 by 10 we get 17 with a remainder so 10 isn't a factor. If we divide 172 by 11 we get 15 with a remainder so 11 isn't a factor. If we divide 172 by 12 we get 14 with a remainder so 12 isn't a factor. If we divide 172 by 13 we get 13 with a remainder so 13 isn't a factor. If we were to divide 172 by any number greater than 13 the result would be less than 13. We've already divided by every whole number less than 13 so we aren't going to find anything new by dividing by numbers greater than 13. Our factors are 2, 86, 4 and 43, as well as 1 and the number 172 itself. A method which is often quicker if the prime factorization contains a large number of factors is to list every prime factor, every product of two prime factors, every product of three prime factors, etc.: From the Prime Factorization 172 = 2 * 2 * 43 you find that the factors include: Each prime factor: 2 and 43 Each product of two prime factors: 2 * 2 = 4 and 2 * 43 = 86 The number itself and 1: 1 and 172. This method is quicker and more reliable than dividing by every possible number (what would you do with 5,668,725, for example?). ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I didn’t put the 86 in there because I just did the prime numbers….. ------------------------------------------------ Self-critique Rating:
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Given Solution: `a** 25025 is not divisible by 2 because it doesn't end in an even number. 25025 isn't divisible by 3 because the sum 2 + 5 + 0 + 2 + 5 = 14 of its digits is not divisible by 3. 25025 isn't divisible by 4 because its last two digits do not form a number divisible by 4. 25025 is divisible by 5 because its last digit is 5. 25025 isn't divisible by 6 because it isn't divisible by 2 and 3. 25025 isn't divisible by 8 because its last three digits do not form a number divisible by 8. 25025 isn't divisible by 9 because the sum 2 + 5 + 0 + 2 + 5 = 14 of its digits is not divisible by 9. 25025 isn't divisible by 12 because it isn't divisible by both 3 and 4.** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Okay I have 5, but I have some others, is it right or did I get it wrong?????? ------------------------------------------------ Self-critique Rating:
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Given Solution: `a** We can follow the simplest method, dividing successively by the smallest possible prime: 360 / 2 = 180, so 2 is a prime factor. 180 / 2 = 90, so 2 is again a factor. 90 / 2 = 45, so 2 is again a factor. 45 can't be divided by 2 so we note that 2 occurs 3 times as a factor and try division by 3: 45 / 3 = 15, so 3 is a factor. 15 / 3 = 5, so 3 is again a factor. 5 is itself prime. It follows that 360 = 2 * 2 * 2 * 3 * 3 * 5, as can be easily checked by multiplication. Thus the prime factorization is 360 = 2^3 * 3^2 * 5. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Did it!!!!! ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q5.1.60 number of divisors of 2^4*3^4*5^2 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 32400 It seems like there is a lot of numbers that are divisors of this number. The first 10 numbers are divisible but 7. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The powers are 4, 4 and 2. The number of possible factors is therefore (4 + 1) * (4 + 1) * (2 + 1) = 5 * 5 * 3 = 75. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Oh geez, I so got that wrong. Don’t know what I was thinking, but I do see what you did. ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qquery 5.1.80 is 2*3*...*13+1 prime? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2*3*5*7*11*13 + 1 = 30031 (yes it is prime) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** To test for primeness you have to divide the number by every prime up to and including its square root. Having done so, you will either find that one of these primes does divide the number, or you will find that none does. Either way you will be able to answer the question. The number we need to test is 2 * 3 * 5 * 7 * 11 * 13 + 1 = 30031. Note that even though this is a pretty good-sized number it's not that big a task to divide by all primes up to the square root. The square root of 30031 is less than 174 so we only have to divide by primes less than 174, and there aren't all that many of them. Besides if one of the numbers 'works' we can stop. In fact 30,031 is not prime. Dividing by the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, all lead to fractional results. But on the next prime 59 we hit paydirt because 30,031 = 59 * 509 so 30031 isn't prime. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qquery 5.1.80 is 2*3*...*13+1 prime? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2*3*5*7*11*13 + 1 = 30031 (yes it is prime) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** To test for primeness you have to divide the number by every prime up to and including its square root. Having done so, you will either find that one of these primes does divide the number, or you will find that none does. Either way you will be able to answer the question. The number we need to test is 2 * 3 * 5 * 7 * 11 * 13 + 1 = 30031. Note that even though this is a pretty good-sized number it's not that big a task to divide by all primes up to the square root. The square root of 30031 is less than 174 so we only have to divide by primes less than 174, and there aren't all that many of them. Besides if one of the numbers 'works' we can stop. In fact 30,031 is not prime. Dividing by the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, all lead to fractional results. But on the next prime 59 we hit paydirt because 30,031 = 59 * 509 so 30031 isn't prime. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!