course Mth 151
......!!!!!!!!...................................
10:55:16 query 5.3.12 using prime factors find the greatest common factor of 180 and 300. What is the greatest common factor and how did you use prime factors to find it?
......!!!!!!!!...................................
RESPONSE --> First find the prime factorization of 180 and 300. 180 2*90 2*2*45 2*2*5*3*3 2^2 x 3^2 x 5 300 2*150 2*2*75 2*2*5*15 2*2*5*3*5 2^2 x 3 x 5^2 Looking at the prime factorizations, pick the numbers that are common between the two and use the smaller exponents. 2^2 x 3 x 5 = 60
.................................................
......!!!!!!!!...................................
10:55:31 ** 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. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
10:57:07 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.
......!!!!!!!!...................................
RESPONSE --> Start with dividing 70 by 25. You get remainder 20. Divide 25 by 20. You get remainder 5. Divide 20 by 5 and you get 0. The last positive remainder was 5, so 5 is the GCF.
.................................................
......!!!!!!!!...................................
10:57:46 ** 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. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
10:59:57 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?
......!!!!!!!!...................................
RESPONSE --> Begin with finding the prime factorizations of 24, 36 and 48. 24 = 2^3 x 3 36 = 2^4 x 3^2 48 = 2^4 x 3 Use the highest exponent for each number. 2^4 x 3^2 = 16 x 9 = 144
.................................................
......!!!!!!!!...................................
11:00:03 ** 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. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
11:03:33 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.
......!!!!!!!!...................................
RESPONSE --> Start with 48 and 315. 315/48 with remainder 27 48/27 with remainder 21 27/21 with remainder 6 21/6 with remainder 3 6/3 with remainder 0 So the GCF between these two numbers is 3. Use 3 and divide that into 450. 450/3 with remainder 0 The GFC for all three numbers is 3.
.................................................
......!!!!!!!!...................................
11:03:46 ** 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. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
11:04:49 Query Add comments on any surprises or insights you experienced as a result of this assignment.
......!!!!!!!!...................................
RESPONSE --> This has been a different review assignment from middle/high school days. I somewhat remember about GCF and LCM but I have never figured them out in these different ways.
.................................................
"