#$&* course mth 151 8-8 7 025. `query 25
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Given Solution: `a** The prime factorizations are 180=2 ^2 * 3 ^ 2 * 5 and 300=2 ^2 * 3 ^1 * 5^2. They have in commin 2^2, 3 and 5, and no higher power of any of these factors. Since 2^2 * 3^1 * 5^1=60 the greatest common factor is 60. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating #$&*:3 ********************************************* Question: `q query 5.3.24 Euclidean algorithm to find GCF(25,70) Show how you used the Euclidean algorithm to find the greatest common factor of the two numbers. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Divide larger number 70 by the smaller number 25 which leaves a remainder of 20, then divide 25 by the remainder 20 and a new remainder appears, 5. Take 20 and divide it by 5, leaving a remainder of 0. The greatest common factor between the two numbers is 5. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** To apply the Euclidean algorithm we divide the larger number by the smaller, obtaining a remainder. We then divide the remainder by the divisor and repeat this process until we get 0 remainder. The greatest common divisor is the last divisor. In this case 70 divided by 25 gives us remainder 20. Then we divide the previous divisor 25 by the remainder 20, obtaining remainder 5. Then we divide the previous divisor, which is now 20, by the remainder 5. The remainder of this division is 0. So the last divisor, which is 5, is the greatest common factor. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating #$&*:3 ********************************************* Question: `q query 5.3.36 LCM of 24, 36, 48 How did you use the prime factors of the given numbers to find their greatest common factor? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 24 = 2*2*2*3 36 = 2*2*3*3 48 = 2*2*2*2*3 By taking the smallest number of factors from the number like, 36 contains the smallest amount of 2, so we take two 2 and multiply them. The next smallest factor amongst the three numbers is 3, 24 has the smallest amount of 3s with one, so add one 3 to the 2*2*3 and 12 is the greatest common factor between these three numbers. Since there are no factors besides 2 and 3, those are the only two numbers we used. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The prime factorizations are 24 = 2*2*2*3, 36 = 2*2*3*3, 48 = 2*2*2*2*3. The smallest number that includes all these factors has four 2's and two 3's. 2*2*2*2 * 3*3 = 144. So 144 is the GCF. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating #$&*:3 ********************************************* Question: `q query 5.3.48 GCF of 48, 315, 450 Show how you used the Euclidean algorithm to find the greatest common factor of the three given numbers. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 48 = 2*2*2*2*3 315 = 3*3*5*7 450 = 2*3*3*5*5 Take 315 and divide it by 48, the remainder is 27. Take 48 and divide it by 27, which leaves a remainder of 21. Now take 27 and divide it by 21, leaving a remainder of 6. By dividing 21 by 6 the remainder will be 3. Finally take 6 and divide it by 3. By looking at the prime factorization of 450, you can gather that the number 3 is the greatest common factor between all three numbers. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** Applying the Euclidean Algorithm to 315 and 48: 315 divided by 48 gives us remainder 27. 48 divided by 27 gives us remainder 21. 27 divided by 21 gives us remainder 3. 6 divided by 3 gives us remainder 0. The last divisor is 3, which is therefore the GCF of 315 and 48. The GCF of the three numbers is therefore the GCF of 450 and 3, which is found by first dividing 450 by 3, which gives us remainder 0. So the last divisor is 3, which is therefore the GCF of the three numbers. ** Query Add comments on any surprises or insights you experienced as a result of this assignment. "