Final answer:
a. The prime factorization of 9 is 3 x 3. The prime factorization of 8 is 2 x 2 x 2. The prime factorization of 6 is 2 x 3. The prime factorization of 0 is not possible because 0 does not have any prime factors. b. The GCD of the integers is 4. c. The LCM of the integers is 72. d. The GCD calculated using the Euclidean algorithm is 3.
Step-by-step explanation:
a. The prime factorization of 9 is 3 x 3. The prime factorization of 8 is 2 x 2 x 2. The prime factorization of 6 is 2 x 3. The prime factorization of 0 is not possible because 0 does not have any prime factors.
b. To find the greatest common divisor (GCD) of the integers, we need to find the common prime factors between them. The common prime factors are 2 x 2 = 4. So, the GCD is 4.
c. To find the least common multiple (LCM) of the integers, we need to find the product of all the prime factors, with the highest exponent for each prime factor. The LCM is 2 x 2 x 2 x 3 x 3 = 72.
d. The Euclidean algorithm is used to find the GCD of two numbers. To apply the Euclidean algorithm, divide the larger number by the smaller number. Keep dividing the remainder by the previous divisor until the remainder is zero. The last divisor used to divide is the GCD. In this case, the larger number is 9 and the smaller number is 6. 9 divided by 6 is 1 with a remainder of 3. Then, 6 divided by 3 is 2 with no remainder. So, the GCD is 3.