2. Write 120 as a product of its prime factors. Show your work for full marks. [
To find the prime factors of 120, we start by finding the smallest prime number that divides 120 evenly, which is 2.
120 ÷ 2 = 60
So we have factored out a 2 and are left with 60. We repeat the same step for 60.
60 ÷ 2 = 30
Again, we have factored out a 2 and are left with 30. We repeat the step for 30.
30 ÷ 2 = 15
We have factored out another 2 and are left with 15. Now we try dividing by the next prime number, which is 3.
15 ÷ 3 = 5
Finally, we have factored out a 3 and are left with 5. At this point, we can see that 5 is a prime number, so we have completed the factoring.
Putting it all together, the prime factors of 120 are 2 x 2 x 2 x 3 x 5 = 2^3 x 3 x 5.
To find the prime factors of 120, we'll start by dividing it by the smallest prime number, which is 2:
120 ÷ 2 = 60
Next, we'll divide 60 by 2 again:
60 ÷ 2 = 30
Now, let's divide 30 by 2 once more:
30 ÷ 2 = 15
Since 15 is an odd number, we'll move on to the next prime number, which is 3. We'll divide 15 by 3:
15 ÷ 3 = 5
5 is a prime number itself, so we cannot divide it any further.
To summarize:
120 = 2 × 2 × 2 × 3 × 5
Therefore, the prime factorization of 120 is 2 × 2 × 2 × 3 × 5.
To find the prime factors of a number, we need to identify the prime numbers that can be multiplied together to obtain the original number.
To write 120 as a product of its prime factors, we follow these steps:
Step 1: Begin by dividing 120 by the smallest prime number, which is 2.
120 ÷ 2 = 60
Step 2: Now, we continue to divide the quotient obtained (60) by 2 until it is no longer divisible evenly.
60 ÷ 2 = 30
30 ÷ 2 = 15
Step 3: Since 15 is not divisible by 2, we proceed to the next prime number, which is 3.
15 ÷ 3 = 5
Step 4: Now, we have obtained 5, which is already a prime number. Since it is not divisible by any prime number greater than itself, we stop here.
Therefore, the prime factorization of 120 is:
2 * 2 * 2 * 3 * 5 = 2^3 * 3 * 5
You can see that 2 has a power of 3 since it appeared 3 times during the factorization process. The remaining prime numbers, 3 and 5, are only there once since they could not be divided any further.
Thus, writing 120 as a product of its prime factors, we have 120 = 2^3 * 3 * 5.