Find the least common multiple of 147 and 63.

To find the least common multiple (LCM) of 147 and 63, follow these steps:

1. Understand the LCM Concept: The least common multiple of two numbers is the smallest positive integer that is divisible by both numbers.

2. Understand the Relationship with GCD: One way to find the LCM is by using the greatest common divisor (GCD) of the numbers. The formula relating LCM and GCD is:
[tex]\[
\text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)}
\][/tex]

3. Find the GCD of 147 and 63:
- Use common factors to determine the GCD:
- The prime factorization of 147 is [tex]$3 \times 7 \times 7$[/tex].
- The prime factorization of 63 is [tex]$3 \times 3 \times 7$[/tex].
- Common factors in both numbers are [tex]$3$[/tex] and [tex]$7$[/tex].
- Multiply the common factors: [tex]$3 \times 7 = 21$[/tex].
- So, the GCD of 147 and 63 is 21.

4. Calculate the LCM Using the Formula:
- Use the formula:
[tex]\[
\text{LCM}(147, 63) = \frac{147 \times 63}{21}
\][/tex]
- Calculate the numerator: [tex]$147 \times 63 = 9261$[/tex].
- Divide by the GCD: [tex]$\frac{9261}{21} = 441$[/tex].

5. Finalize the Result:
- The least common multiple of 147 and 63 is 441.

Therefore, the LCM of 147 and 63 is [tex]$441$[/tex].