Question

1. write each number as a product of prime numbers: (a) 360 (b) 720 (c) 144\
2. list all the common factors of each pair of numbers: 15 and 18\
3. 36 and 45\
4. 9 and 10

Explanation:

Problem 13 (Prime Factorization)

Part (a): 360

Step 1: Divide by 2

Start with the smallest prime number, 2. \( 360 \div 2 = 180 \)

Step 2: Divide by 2 again

\( 180 \div 2 = 90 \)

Step 3: Divide by 2 again

\( 90 \div 2 = 45 \)

Step 4: Divide by 3

Now, divide by the next prime number, 3. \( 45 \div 3 = 15 \)

Step 5: Divide by 3 again

\( 15 \div 3 = 5 \)

Step 6: Divide by 5

Finally, divide by 5. \( 5 \div 5 = 1 \)
So, \( 360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5 \)

Part (b): 720

Step 1: Divide by 2

\( 720 \div 2 = 360 \)

Step 2: Divide by 2 again

\( 360 \div 2 = 180 \)

Step 3: Divide by 2 again

\( 180 \div 2 = 90 \)

Step 4: Divide by 2 again

\( 90 \div 2 = 45 \)

Step 5: Divide by 3

\( 45 \div 3 = 15 \)

Step 6: Divide by 3 again

\( 15 \div 3 = 5 \)

Step 7: Divide by 5

\( 5 \div 5 = 1 \)
So, \( 720 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^4 \times 3^2 \times 5 \)

Part (c): 144

Step 1: Divide by 2

\( 144 \div 2 = 72 \)

Step 2: Divide by 2 again

\( 72 \div 2 = 36 \)

Step 3: Divide by 2 again

\( 36 \div 2 = 18 \)

Step 4: Divide by 2 again

\( 18 \div 2 = 9 \)

Step 5: Divide by 3

\( 9 \div 3 = 3 \)

Step 6: Divide by 3 again

\( 3 \div 3 = 1 \)
So, \( 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2 \)

Problem 14: Common Factors of 15 and 18

Answer:

Step 1: Find factors of 9

The factors of 9 are \( 1, 3, 9 \) (since \( 1 \times 9 = 9 \), \( 3 \times 3 = 9 \))

Step 2: Find factors of 10

The factors of 10 are \( 1, 2, 5, 10 \) (since \( 1 \times 10 = 10 \), \( 2 \times 5 = 10 \))

Step 3: Identify common factors

The only common factor is \( 1 \) (since 1 is a factor of every number)