Find the LCM of 96 and 360 using fundamental theorem of Arithmetic.

Answer

To find the LCM (Least Common Multiple) of 96 and 360 using the Fundamental Theorem of Arithmetic, we will follow these steps:

### Step 1: Prime Factorization of 96
First, we need to find the prime factorization of 96.

- Divide 96 by 2:
- \( 96 \div 2 = 48 \)
- Divide 48 by 2:
- \( 48 \div 2 = 24 \)
- Divide 24 by 2:
- \( 24 \div 2 = 12 \)
- Divide 12 by 2:
- \( 12 \div 2 = 6 \)
- Divide 6 by 2:
- \( 6 \div 2 = 3 \)
- Finally, divide 3 by 3:
- \( 3 \div 3 = 1 \)

So, the prime factorization of 96 is:
\[ 96 = 2^5 \times 3^1 \]

### Step 2: Prime Factorization of 360
Next, we will find the prime factorization of 360.

- Divide 360 by 2:
- \( 360 \div 2 = 180 \)
- Divide 180 by 2:
- \( 180 \div 2 = 90 \)
- Divide 90 by 2:
- \( 90 \div 2 = 45 \)
- Divide 45 by 3:
- \( 45 \div 3 = 15 \)
- Divide 15 by 3:
- \( 15 \div 3 = 5 \)
- Finally, divide 5 by 5:
- \( 5 \div 5 = 1 \)

So, the prime factorization of 360 is:
\[ 360 = 2^3 \times 3^2 \times 5^1 \]

### Step 3: Identify the Highest Powers of Each Prime Factor
Now we will identify the highest powers of all prime factors from both factorizations.

- For the prime factor 2:
- The highest power is \( 2^5 \) (from 96).
- For the prime factor 3:
- The highest power is \( 3^2 \) (from 360).
- For the prime factor 5:
- The highest power is \( 5^1 \) (from 360).

### Step 4: Calculate the LCM
Now we can calculate the LCM by multiplying the highest powers of each prime factor together:

\[ \text{LCM} = 2^5 \times 3^2 \times 5^1 \]

Calculating this step-by-step:

1. Calculate \( 2^5 = 32 \)
2. Calculate \( 3^2 = 9 \)
3. Calculate \( 5^1 = 5 \)

Now multiply these results together:

\[ \text{LCM} = 32 \times 9 \times 5 \]

Calculating \( 32 \times 9 \):

\[ 32 \times 9 = 288 \]

Now multiply \( 288 \times 5 \):

\[ 288 \times 5 = 1440 \]

### Final Answer
Thus, the LCM of 96 and 360 is:
\[ \text{LCM} = 1440 \]

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