Final answer:
a) The prime factorization of 600 is 2³ × 3 × 5. b) The highest common factor of 600 and 1059 is 3.
Step-by-step explanation:
a) To find the prime factorization of 600, we start by dividing it by the smallest prime number, which is 2.
600 ÷ 2 = 300
Now, we divide 300 by 2 again:
300 ÷ 2 = 150
Next, we divide 150 by 2:
150 ÷ 2 = 75
We can't divide 75 by 2 anymore, so we move on to the next prime number, which is 3:
75 ÷ 3 = 25
Finally, we divide 25 by 5:
25 ÷ 5 = 5
So the prime factorization of 600 is 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5.
b) To find the highest common factor of 600 and 1059, we need to find the common prime factors and multiply them together. From part a), we know the prime factorization of 600 is 2³ × 3 × 5. Now let's find the prime factorization of 1059:
1059 is not divisible by 2, so we try 3:
1059 ÷ 3 = 353
353 is a prime number, so the prime factorization of 1059 is 3 × 353.
The common prime factor between 600 and 1059 is 3, so the highest common factor is 3.