Write each number as the product of prime factors.
11. 110
13.180
15.441
17. 1080
11. 110 = 2 × 5 × 11
13. 180 = 2 × 2 × 3 × 3 × 5
15. 441 = 3 × 3 × 7 × 7
17. 1080 = 2 × 2 × 2 × 3 × 3 × 3 × 5
Sure! Here are the step-by-step explanations for each number:
11. 110:
To express 110 as the product of prime factors, we need to find prime numbers that can divide 110 without a remainder.
Step 1: Start by dividing 110 by the smallest prime number, which is 2:
110 ÷ 2 = 55
Step 2: Now, we have 55. Let's divide it again by 2:
55 ÷ 2 = Not divisible by 2
Step 3: Next, divide 55 by the next prime number, which is 3:
55 ÷ 3 = Not divisible by 3
Step 4: Since 55 is not divisible by 2 or 3, we need to try with the next prime number, which is 5:
55 ÷ 5 = 11
Step 5: Finally, we have 11 left, which is already a prime number. Therefore, the prime factorization of 110 is:
2 × 5 × 11 = 110
13. 180:
To express 180 as the product of prime factors, let's go through the following steps:
Step 1: Divide 180 by the smallest prime number, which is 2:
180 ÷ 2 = 90
Step 2: Now, divide 90 by 2 again:
90 ÷ 2 = 45
Step 3: Further divide 45 by 3:
45 ÷ 3 = 15
Step 4: We are left with 15, which is divisible by 3:
15 ÷ 3 = 5
Step 5: 5 is a prime number. Therefore, the prime factorization of 180 is:
2 × 2 × 3 × 3 × 5 = 180
15. 441:
To express 441 as the product of prime factors, follow these steps:
Step 1: Divide 441 by the smallest prime number, which is 3:
441 ÷ 3 = 147
Step 2: Now, divide 147 by 3 again:
147 ÷ 3 = 49
Step 3: Further divide 49 by 7:
49 ÷ 7 = 7
Step 4: 7 is a prime number itself, so we are done. The prime factorization of 441 is:
3 × 3 × 7 × 7 = 441
17. 1080:
To express 1080 as the product of prime factors, let's go through these steps:
Step 1: Divide 1080 by the smallest prime number, which is 2:
1080 ÷ 2 = 540
Step 2: Now, divide 540 by 2 again:
540 ÷ 2 = 270
Step 3: Further divide 270 by 2:
270 ÷ 2 = 135
Step 4: Next, divide 135 by 3:
135 ÷ 3 = 45
Step 5: Divide 45 by 3 again:
45 ÷ 3 = 15
Step 6: Finally, divide 15 by 5:
15 ÷ 5 = 3
Step 7: 3 is a prime number. Therefore, the prime factorization of 1080 is:
2 × 2 × 2 × 3 × 3 × 3 × 5 = 1080
To write each number as the product of prime factors, we need to find the prime numbers that divide these numbers evenly. Here's how you can find the prime factors for each given number:
11. 110:
To find the prime factors of 110, we start by dividing it by the smallest prime number, which is 2. Since 110 is an even number, it can be divided evenly by 2. Hence, the prime factorization of 110 is 2 × 55.
55 can further be divided by 5, another prime number. So, the prime factorization of 110 is 2 × 5 × 11.
Therefore, the prime factorization of 110 is 2 × 5 × 11.
13. 180:
To find the prime factors of 180, we will start by dividing it by the smallest prime number, which is 2. 180 is an even number, so it can be divided evenly by 2. Hence, the prime factorization of 180 is 2 × 90.
Next, we divide 90 by 2 again, resulting in 2 × 45.
45 can be divided evenly by 3, another prime number. So, the prime factorization of 180 is 2 × 2 × 3 × 3 × 5.
Therefore, the prime factorization of 180 is 2 × 2 × 3 × 3 × 5.
15. 441:
To find the prime factors of 441, we will start by dividing it by the smallest prime number, which is 3. 441 is divisible by 3, resulting in 3 × 147.
Next, we divide 147 by 3 again, resulting in 3 × 49.
49 can further be divided by 7, another prime number. So, the prime factorization of 441 is 3 × 3 × 7 × 7.
Therefore, the prime factorization of 441 is 3 × 3 × 7 × 7.
17. 1080:
To find the prime factors of 1080, we start by dividing it by the smallest prime number, which is 2. Since 1080 is an even number, it can be divided evenly by 2. Hence, the prime factorization of 1080 is 2 × 540.
Next, we divide 540 by 2 again, resulting in 2 × 270.
270 can be divided evenly by 2: 2 × 135.
135 can be divided evenly by 3: 3 × 45.
45 can further be divided by 3: 3 × 3 × 5.
Therefore, the prime factorization of 1080 is 2 × 2 × 2 × 3 × 3 × 3 × 5.