Find the LCM of 63 and 140 .

Options

A. 63
B. 14
C. 126
D. 1260

Answer

To find the Least Common Multiple (LCM) of 63 and 140, we will follow these steps:

### Step 1: Prime Factorization of 63
- We start by dividing 63 by the smallest prime number, which is 3.
- \( 63 \div 3 = 21 \)
- Next, we divide 21 by 3 again.
- \( 21 \div 3 = 7 \)
- Finally, 7 is a prime number.
- So, the prime factorization of 63 is:
\[ 63 = 3^2 \times 7^1 \]

### Step 2: Prime Factorization of 140
- We start by dividing 140 by the smallest prime number, which is 2.
- \( 140 \div 2 = 70 \)
- Next, we divide 70 by 2 again.
- \( 70 \div 2 = 35 \)
- Now, we divide 35 by the next smallest prime number, which is 5.
- \( 35 \div 5 = 7 \)
- Finally, 7 is a prime number.
- So, the prime factorization of 140 is:
\[ 140 = 2^2 \times 5^1 \times 7^1 \]

### Step 3: Identify the Highest Powers of Each Prime
- Now, we will identify the highest power of each prime factor from both factorizations:
- For the prime number 2: The highest power is \( 2^2 \) (from 140).
- For the prime number 3: The highest power is \( 3^2 \) (from 63).
- For the prime number 5: The highest power is \( 5^1 \) (from 140).
- For the prime number 7: The highest power is \( 7^1 \) (from both).

### Step 4: Calculate the LCM
- We multiply the highest powers of all prime factors together to find the LCM:
\[ \text{LCM} = 2^2 \times 3^2 \times 5^1 \times 7^1 \]
- Calculating this step-by-step:
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 5^1 = 5 \)
- \( 7^1 = 7 \)

Now, we multiply these values:
\[ 4 \times 9 = 36 \]
\[ 36 \times 5 = 180 \]
\[ 180 \times 7 = 1260 \]

### Final Answer
The LCM of 63 and 140 is:
\[ \text{LCM} = 1260 \]